I use a-lists all the time.
When and why do you (or should you) use hash tables?
My reluctance to using them is that, unlike the other data structures, hashtables in CL are not visible lists. Which honestly, I find weird considering almost everything is a list.
Maybe I am missing something in my inexperience?
There are lot of assumptions to address in your question:
I believe common lisp is the only language I have worked with that have a variety of extremely useful data structures.
I don't think this is particularly true, the standard libraries of popular languages are filled with lot of data structures too (C++, Java, Rust, Python)
When and why do you (or should you) use hash tables?
Data-structures come with costs in terms of memory and processor usage: a list must be searched linearly to find an element, whereas an hash-table has a constant lookup cost: for small lists however the linear search might be faster than the constant lookup. Moreover, there are other criteria like: do I want to access the data concurrently? a List can be manipulated in a purely functional way, making data-sharing across threads easier than with a hash-table (but hash-table can be associated with a mutex, etc.)
My reluctance to using them is that, unlike the other data structures, hashtables in CL are not visible lists. Which honestly, I find weird considering almost everything is a list.
The source code of Lisp programs is made mostly of Lists and symbols, even if there is no such restriction. But at runtime, CL has a lot of different types that are not at all related to lists: bignums, floating points, rational numbers, complex numbers, vectors, arrays, packages, symbols, strings, classes and structures, hash-tables, readtables, functions, etc. You can model a lot of data at runtime by putting them in lists, which is something that works well for a lot of cases, but they are by far not the only types available.
Just to emphasize a little bit, when you write:
(vector 0 1 2)
This might look like a list in your code, but at runtime the value really is a different kind of object, a vector. Do not be confused by how things are expressed in code and how they are represented during code execution.
If you don't use it already, I suggest installing and using the Alexandria Lisp libray (see https://alexandria.common-lisp.dev/). There are useful functions to convert from and to hash-tables from alists or plists.
More generally, I think it is important to architecture your libraries and programs in a way that hide implementation details: you define a function make-person and accessors person-age, person-name, etc. as well as other user-facing functions. And the actual implementation can use hash tables, lists, etc. but this is not really a concern that should be exposed, because exposing that is a risk: you won't be able to easily change your mind later if you find out that the performance is bad or if you want to add a cache, use a database, etc.
I find however that CL is good at making nice interfaces that do not come with too much accidental complexity.