I don't really have a great feeling for the information captured by the "toposophic" fundamental group so this answer is a bit one-sided towards what I can say about the quotient topology on the fundamental group and what it has to do with covering maps. Maybe you can form something useful from this.

First, I would say $\pi_{1}(X)$ with the quotient topology is "rarely" a topological group but is always "quasi"topological group (in that you know multiplication is separately continuous). It comes with some serious topological baggage. You can fully describe the quotient topology in terms of open covers of the loop space but not usually coverings of $X$ itself. This description in terms of coverings helps with the intuition of what open sets look like but is possible for arbitrary quotient topologies and is not very practical. The requirement that $X$ be "locally simply connected" does rule out a good number of "wild" spaces of interest but there are some very interesting locally simply connected, non-locally path connected examples.

The connection to coverings is: If $p:Y\rightarrow X$ is a covering map, the induced homomorphism $p_{\ast}:\pi_{1}(Y)\rightarrow \pi_{1}(X)$ is an open embedding of quasitopological groups.

This also implies that if $X$ is nice enough to have a universal cover, then $\pi_{1}(X)$ is a discrete group and the topology gives no new information.

Constructing covers of the non-locally path connected spaces you are interested in is tricky so I am not completely sure if there is a complete Galois correspondence between open subgroups and covers. It seems like this would work in the locally path connected case.

I don't really have a great feeling for the information captured by the "toposophic" fundamental group so this answer is a bit one-sided towards what I can say about the quotient topology on the fundamental group and what it has to do with covering maps. Maybe you can form something useful from this.

First, I would say quotient $\pi_{1}(X)$ is "rarely" a topological group but is always "quasi"topological group in that you know multiplication is separately continuous. It comes with some serious topological baggage. The requirement that $X$ be locally simply connected (in the sense I define in the comments) does rule out a good number of "wild" spaces of interest but there are some very interesting locally simply connected, non-locally path connected examples.

The connection to coverings is: If $p:Y\rightarrow X$ is a covering map, the induced homomorphism $p_{\ast}:\pi_{1}(Y)\rightarrow \pi_{1}(X)$ is an open embedding of quasitopological groups.

This also implies that if $X$ is nice enough (for instance, locally simply connectivity in the sense of Wikipedia) to have a universal cover, then $\pi_{1}(X)$ is a discrete group and the topology gives no new information.

Constructing covers of non-locally path connected spaces is tricky so in general there will not be a complete Galois correspondence between open subgroups and covers. There should be a correspondence in the locally path connected case.

Added:

An enlightening difference between quotient and inverse limit topology is in the case of the Hawaiian earring.

It has been known for less than two years that quotient $\pi_1$ is not always a topological group. Besides that, its theory is not very developed. I suppose it is "wrong" if you demand a functor that takes values in the category topological groups. But as long as we are discussing other potential topologies...

There is a natural "fix" to the quotient topology using a reflection from topological algebra. It is defined in The fundamental group as topological group on my personal page. The resulting topologized fundamental group is defined for all spaces, takes values in topological groups, and is universal with respect to continuous homomorphisms from quotient $\pi_1$ to topological groups. It is well-behaved in that it admits a number of topological analogues: On "non-discrete wedges of circles" $X_+\wedge S^1$ it is free topological, every topological group is realized as a fundamental group with this topology by attaching 2-cells to one of these "wedges," van Kampen theorems involving pushouts of topological groups are possible, etc.

I don't really have a great feeling for the information captured by the "toposophic" fundamental group so this answer is a bit one-sided towards what I can say about the quotient topology on the fundamental group and what it has to do with covering maps. Maybe you can form something useful from this.

First, I would say $\pi_{1}(X)$ with the quotient topology is "rarely" a topological group but is always "quasi"topological group (in that you know multiplication is separately continuous). It comes with some serious topological baggage. You can fully describe the quotient topology in terms of coverings of the loop space but not usually coverings of $X$ itself. This description in terms of coverings helps with the intuition of what open sets look like but is possible for arbitrary quotient topologies and is not very practical. The requirement that $X$ be "locally simply connected" does rule out a good number of "wild" spaces of interest but there are some very interesting locally simply connected, non-locally path connected examples.

The connection to coverings is: If $p:Y\rightarrow X$ is a covering map, the induced homomorphism $p_{\ast}:\pi_{1}(Y)\rightarrow \pi_{1}(X)$ is an open embedding of quasitopological groups.

This also implies that if $X$ is nice enough to have a universal cover, then $\pi_{1}(X)$ is a discrete group and the topology gives no new information.

Constructing covers of the non-locally path connected spaces you are interested in is tricky so I am not completely sure if there is a complete Galois correspondence between open subgroups and covers. It seems like this would work in the locally path connected case.

I don't really have a great feeling for the information captured by the "toposophic" fundamental group so this answer is a bit one-sided towards what I can say about the quotient topology on the fundamental group and what it has to do with covering maps. Maybe you can form something useful from this.

First, I would say $\pi_{1}(X)$ with the quotient topology is "rarely" a topological group but is always "quasi"topological group (in that you know multiplication is separately continuous). It comes with some serious topological baggage. You can fully describe the quotient topology in terms of open covers of the loop space but not usually coverings of $X$ itself. This description in terms of coverings helps with the intuition of what open sets look like but is possible for arbitrary quotient topologies and is not very practical. The requirement that $X$ be "locally simply connected" does rule out a good number of "wild" spaces of interest but there are some very interesting locally simply connected, non-locally path connected examples.

The connection to coverings is: If $p:Y\rightarrow X$ is a covering map, the induced homomorphism $p_{\ast}:\pi_{1}(Y)\rightarrow \pi_{1}(X)$ is an open embedding of quasitopological groups.

This also implies that if $X$ is nice enough to have a universal cover, then $\pi_{1}(X)$ is a discrete group and the topology gives no new information.

Constructing covers of the non-locally path connected spaces you are interested in is tricky so I am not completely sure if there is a complete Galois correspondence between open subgroups and covers. It seems like this would work in the locally path connected case.