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.