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Remark. I think a philosophical point of view would help illustrate the difficulty. Note that in general $Homeo(X,Y)$ fails to be closed in $C(X,Y)$ -- the space of continuous function of $X$ to $Y$. An arbitrary Cauchy sequence of homeomorphisms in $Homeo(X,Y)$ may not converge to a homeomorphism, the injectivity is an issue. A commonly used approach to extend the homeomorphism to the limit is when constructing a sequence of homeomorphisms $h_k:X\to Y$ recursively, one need to impose control limiting the distance between $h_k$ and $h_{k+1}$ after $h_1,\dots,h_k$ are all specified. Because of the wildness of knots like the Bing sling, it's not easy to place such a desired control. Ancel's argument doesn't tell us how to find the next $h_{k+1}$.

First, I'd like to make some conventions. Knots $J$ and $K$ are non-ambiently isotopic if there is a level-preserving embedding $e:J \times [0,1]\to S^3 \times [0,1]$ such that $e(J \times \{0\}) = J \times \{0\}$ and $e(J \times \{1\}) = K \times \{1\}$. I think that's what OP means for topologically isotopy of embeddings. For historical reason, I prefer to call it non-ambient isotopy to distinguish from the one that knot theorists study the most.

A simple closed curve $J$ is said to pierce a disk $D$ if $J$ links $Bd D$ and $J \cap D$ is a single point.

Fact. Every knot that pierces a tame disk is non-ambiently isotopic to an unknot.

I stole the pictorial proof from Ric Ancel's preprint.

The Bing sling is a wild knot that pierces no disk and every point of it is wild. The following picture is quoted from P. 81 in Daverman-Venema's book, a similar one may find in Ancel's preprint as well. The Bing sling is the intersection of the nested tori. Although the Bing sling (denoted by) $\Sigma$ looks intimidating, it would be a fun exercise to show that the Bing sling (the innermost curve) is homeomorphic to the core $J$ of $T_1$. Or one may reembed the $J$ by infinitely many times so the limit is the Bing sling.

A big open question is whether every knot is non-ambiently isotopic to an unknot. (I'm not sure if it was first due to Rolfsen). In addition, it is unknown if the Bing sling is non-ambiently isotopic to an unknot.

If I understand correctly, the approach described in OP about squeezing $T_1$ towards $\Sigma$ so there is a sequence of successive homeomorphisms sending the core $J$ of $T_1$ onto $\Sigma$ is carried out in Ancel's preprint. Actually, he proved something more general but unfortunately, a partial result to what OP hoped.

Theorem. Every knot is semi-isotopic to an unknot.

A knot $J$ is semi-isotopic to a knot $K$ if there is annulus $A$ is $S^3 \times [0,1]$ such that $\partial A = (J \times \{0\}) \cup (K \times \{1\})$ and there is a homeomorphism $e: S^1 \times [0,1) \to A - (K \times \{1\})$ such that $e(S^1 \times \{t\}) \subset S^3 \times \{t\}$ for every $t \in [0,1)$.
However, $e$ may not extend continuously to homeomorphism from $S^1 \times [0,1]$ onto $A$. I don't know how to push this idea further because knots like Bing sling are everywhere wild, one can't just contract the wild portion to a point. Ancel considered a mapping cylinder between $J$ and $K$ and showed that one can level up the wild points in a "swirling way" and then comb the points towards $K$ but it's unclear what will happen at the end $K$.

First, I'd like to make some conventions. Knots $J$ and $K$ are non-ambiently isotopic if there is a level-preserving embedding $e:J \times [0,1]\to S^3 \times [0,1]$ such that $e(J \times \{0\}) = J \times \{0\}$ and $e(J \times \{1\}) = K \times \{1\}$. I think that's what OP means for topologically isotopy of embeddings. For historical reason, I prefer to call it non-ambient isotopy to distinguish from the one that knot theorists study the most.

A simple closed curve $J$ is said to pierce a disk $D$ if $J$ links $Bd D$ and $J \cap D$ is a single point.

Fact. Every knot that pierces a tame disk is non-ambiently isotopic to an unknot.

I stole the pictorial proof from Ric Ancel's preprint.

The Bing sling is a wild knot that pierces no disk and every point of it is wild. The following picture is quoted from P. 81 in Daverman-Venema's book, a similar one may find in Ancel's preprint as well. The Bing sling is the intersection of the nested tori. Although the Bing sling (denoted by) $\Sigma$ looks intimidating, it would be a fun exercise to show that the Bing sling (the innermost curve) is homeomorphic to the core $J$ of $T_1$. Or one may reembed the $J$ by infinitely many times so the limit is the Bing sling.

A big open question is whether every knot is non-ambiently isotopic to an unknot. (I'm not sure if it was first due to Rolfsen). In addition, it is unknown if the Bing sling is non-ambiently isotopic to an unknot.

If I understand correctly, the approach described in OP about squeezing $T_1$ towards $\Sigma$ so there is a sequence of successive homeomorphisms sending the core $J$ of $T_1$ onto $\Sigma$ is carried out in Ancel's preprint. Actually, he proved something more general but unfortunately, a partial result to what OP hoped.

Theorem. Every knot is semi-isotopic to an unknot.

A knot $J$ is semi-isotopic to a knot $K$ if there is annulus $A$ is $S^3 \times [0,1]$ such that $\partial A = (J \times \{0\}) \cup (K \times \{1\})$ and there is a homeomorphism $e: S^1 \times [0,1) \to A - (K \times \{1\})$ such that $e(S^1 \times \{t\}) \subset S^3 \times \{t\}$ for every $t \in [0,1)$.
However, $e$ may not extend continuously to homeomorphism from $S^1 \times [0,1]$ onto $A$. I don't know how to push this idea further because knots like Bing sling are everywhere wild, one can't just contract the wild portion to a point. A brief sketch of Ancel's argument is the following: Every knot $J$ can be considered as a subset of the interior of an unknotted solid torus $T$. Let $K$ be the core of $T$. Then there is an obvious map $\pi: T \to K$ by squeezing $T$ towards the core. Since $\pi|J: J \to K$ is a homotopy equivalence, $\pi|J: J \to K$ is homotopic to an homeomorphism $\chi: J \to K$. Using the swirling map, one may build a mapping swirl $Swl(\pi|J)$ between $J$ and $K$. Roughly speaking, it's like a twisted mapping cylinder. It turns out that $Swl(\pi|J)$ is homeomorphic to a mapping cylinder $Cyl(\chi)$. Therefore, $Swl(\pi|J)$ minus the right end is homeomorphic to an annulus $A - (K \times \{1\})$, and every level of it is a homeomorphism. So things before $t =1$ can be lined up pretty nicely (in my view, it's like "combing" towards $K$), but it's unclear what will happen at the end $K$ in the mapping cylinder.

First, I'd like to make some conventions. Knots $J$ and $K$ are non-ambiently isotopic if there is a level-preserving embedding $e:J \times [0,1]\to S^3 \times [0,1]$ such that $e(J \times \{0\}) = J \times \{0\}$ and $e(J \times \{1\}) = K \times \{1\}$. I think that's what OP means for topologically isotopy of embeddings. For historical reason, I prefer to call it non-ambient isotopy to distinguish from the one that knot theorists study the most.

A simple closed curve $J$ is said to pierce a disk $D$ if $J$ links $Bd D$ and $J \cap D$ is a single point.

Fact. Every knot that pierces a tame disk is non-ambiently isotopic to an unknot.

I stole the pictorial proof from Ric Ancel's preprint.

The Bing sling is a wild knot that pierces no disk and every point of it is wild. The following picture is quoted from Daverman-Venema's book, a similar one may find in Ancel's preprint as well. The Bing sling is the intersection of the nested tori. Although the Bing sling (denoted by) $\Sigma$ looks intimidating, it would be a fun exercise to show that the Bing sling (the innermost curve) is homeomorphic to the core $J$ of $T_1$. Or one may reembed the $J$ by infinitely many times so the limit is the Bing sling.

A big open question is whether every knot is non-ambiently isotopic to an unknot. (I'm not sure if it was first due to Rolfsen). In addition, it is unknown if the Bing sling is non-ambiently isotopic to an unknot.

If I understand correctly, the approach described in OP about squeezing $T_1$ towards $\Sigma$ so there is a sequence of successive homeomorphisms sending the core $J$ of $T_1$ onto $\Sigma$ is carried out in Ancel's preprint. Actually, he proved something more general but unfortunately, a partial result to what OP hoped.

Theorem. Every knot is semi-isotopic to an unknot.

A knot $J$ is semi-isotopic to a knot $K$ if there is annulus $A$ is $S^3 \times [0,1]$ such that $\partial A = (J \times \{0\}) \cup (K \times \{1\})$ and there is a homeomorphism $e: S^1 \times [0,1) \to A - (K \times \{1\})$ such that $e(S^1 \times \{t\}) \subset S^3 \times \{t\}$ for every $t \in [0,1)$.
However, $e$ may not extend continuously to homeomorphism from $S^1 \times [0,1]$ onto $A$. I don't know how to push this idea further because knots like Bing sling are everywhere wild, one can't just contract the wild portion to a point. Ancel considered a mapping cylinder between $J$ and $K$ and showed that one can level up the wild points in a "swirling way" and then comb the points towards $K$ but it's unclear what will happen at the end $K$.

First, I'd like to make some conventions. Knots $J$ and $K$ are non-ambiently isotopic if there is a level-preserving embedding $e:J \times [0,1]\to S^3 \times [0,1]$ such that $e(J \times \{0\}) = J \times \{0\}$ and $e(J \times \{1\}) = K \times \{1\}$. I think that's what OP means for topologically isotopy of embeddings. For historical reason, I prefer to call it non-ambient isotopy to distinguish from the one that knot theorists study the most.

A simple closed curve $J$ is said to pierce a disk $D$ if $J$ links $Bd D$ and $J \cap D$ is a single point.

Fact. Every knot that pierces a tame disk is non-ambiently isotopic to an unknot.

I stole the pictorial proof from Ric Ancel's preprint.

The Bing sling is a wild knot that pierces no disk and every point of it is wild. The following picture is quoted from P. 81 in Daverman-Venema's book, a similar one may find in Ancel's preprint as well. The Bing sling is the intersection of the nested tori. Although the Bing sling (denoted by) $\Sigma$ looks intimidating, it would be a fun exercise to show that the Bing sling (the innermost curve) is homeomorphic to the core $J$ of $T_1$. Or one may reembed the $J$ by infinitely many times so the limit is the Bing sling.

A big open question is whether every knot is non-ambiently isotopic to an unknot. (I'm not sure if it was first due to Rolfsen). In addition, it is unknown if the Bing sling is non-ambiently isotopic to an unknot.

If I understand correctly, the approach described in OP about squeezing $T_1$ towards $\Sigma$ so there is a sequence of successive homeomorphisms sending the core $J$ of $T_1$ onto $\Sigma$ is carried out in Ancel's preprint. Actually, he proved something more general but unfortunately, a partial result to what OP hoped.

Theorem. Every knot is semi-isotopic to an unknot.

A knot $J$ is semi-isotopic to a knot $K$ if there is annulus $A$ is $S^3 \times [0,1]$ such that $\partial A = (J \times \{0\}) \cup (K \times \{1\})$ and there is a homeomorphism $e: S^1 \times [0,1) \to A - (K \times \{1\})$ such that $e(S^1 \times \{t\}) \subset S^3 \times \{t\}$ for every $t \in [0,1)$.
However, $e$ may not extend continuously to homeomorphism from $S^1 \times [0,1]$ onto $A$. I don't know how to push this idea further because knots like Bing sling are everywhere wild, one can't just contract the wild portion to a point. Ancel considered a mapping cylinder between $J$ and $K$ and showed that one can level up the wild points in a "swirling way" and then comb the points towards $K$ but it's unclear what will happen at the end $K$.