Here's a scheme to produce slender (noetherian) 2-generated groups that contain all finite subgroups (and quasi-finite, in the sense that all proper subgroups are finite).

Given a group $G$, let $C(G)$ be the set of conjugacy classes of finite subgroups of $G$. A group homomorphism $f:G\to H$ induces a map $f_*:C(G)\to C(H)$. Let us say that $f$ is safe

- $f_*$ is injective
- for every $P\in C(G)\smallsetminus\{\{1\}\}$, $f(P)$ is contained in no $P'\in C(H)\smallsetminus f_*(C(G))$

We say that $Q$ is a safe quotient of $G$ if the projection $G\to Q$ is safe.

Remark: $f$ safe implies that $f$ is injective in restriction to any finite subgroup: indeed the first condition is that non-conjugate finite subgroups map to non-conjugate subgroups; applying this to $\{1\}$ and the cyclic subgroup generated by some element of finite order, we conclude.

The second condition means, in rough terms, that a finite subgroup of $G$ cannot be embedded in a larger finite subgroup in $H$, except in a way that comes from $G$.

"Conjecture": let $G$ be a non-elementary hyperbolic group.

- If $H$ is a non-elementary subgroup, there exists a safe non-elementary hyperbolic quotient of $G$ such that the image of $H$ is all of $Q$.
- If $H$ is a virtually infinite cyclic subgroup, there exists a safe non-elementary hyperbolic quotient of $G$ such that the image of $H$ is finite
- If $F$ is any finite group, there exists a safe non-elementary hyperbolic quotient of $G$ containing a copy of $F$.

Statements close to (1) and (2) are stated in Gromov's original book and probably follow rigorously from later work, notably by Olshanski or Delzant. I'll update if I get more precise references in the comments. I guess (3) is of similar difficulty; the issue here is this safeness condition; however it's very plausible that it follows from the same methods.

Corollary of the "conjecture": every non-elementary hyperbolic group $G$ has a quotient in which all proper subgroups are finite, and containing isomorphic copy of all finite groups.

Proof of the corollary: enumerate the finitely generated subgroups of $G$ as ($L_n$). Define $G_0=G$. If $G_n$ is defined with a quotient map $G\to G_n$ and $G_n$ is hyperbolic, let $H_n$ be the image of $L_n$ in $G_n$. If $H_n$. Using (1) or (2), define a safe non-elementary hyperbolic quotient $G'_n$ of $G_n$ in which the image of $G_n$ is either finite or all of $G'_n$ (if $G_n$ is finite just take $G'_n=G_n$). Next, using (3), define a safe non-elementary hyperbolic quotient $G_{n+1}$ of $G'_n$ containing a copy of the symmetric group $S_n$.

Define $G_\infty$ as the inductive limit. Clearly it contains copies of all finite subgroups (as the successive maps are injective on finite subgroups). First let $H$ be a proper finitely generated subgroup. So $H$ is the image of $L_n$ for some $n$. Then the image of $L_n$ in $G_{n+1}$ is either finite or all of $G_{n+1}$, and hence $H$ is either finite or all of $G_\infty$. Finally, we have to prove that $G$ has no infinite locally finite subgroup. Indeed, suppose that $M$ is one. Let $M_1$ be some nontrivial finite subgroup in $M_1$; it can be lifted to an isomorphic copy $M'_1$ of itself in $G_n$ for large enough $n$. We fix such $n$. Let $W\subset G_n$ be the inverse image of $M$ in $G_n$ and let $M'_2$ be a maximal finite subgroup of $W$ containing $M'_1$, and $M_2$ its image in $G_\infty$. Let $M_3$ be a finite subgroup properly containing $M_2$ in $M$; it can be lifted in $G_m$ for some $m>n$, to a subgroup $M''_3$. Then the image of $M_2$ in $G_m$ is properly contained in $M''_3$. This contradicts the safeness assumption.