The role of ANR in modern topology

Another reason you might not see the word ANR these days is that compact finite-dimensional spaces are ANRs if and only if they are locally contractible. Thus, "finite-dimensional and local contractible" can replace ANR in the statement of a theorem (and might help the result appeal to a wider audience).

In comparison geometry, for instance, the existence of a contractibility function takes the place of the ANR condition.

Borsuk conjectured that compact ANRs should have the homotopy types of finite simplicial complexes. Chapman and West proved that they even have preferred simple-homotopy types. This is part of the "topological invariance of torsion" package and is quite a striking result. Every compact, finite-dimensional, locally contractible space has a preferred finite combinatorial structure that is well-defined up to (even local!) simple-homotopy moves.

ANRs are (and have always been) irrelevant as long as homotopy-invariant properties of spaces homotopy equivalent to CW-complexes are concerned. But modern algebraic topologists do not seem to be really interested in (or anyway have real tools to deal with) more general spaces AFAIK. (Of course, "general nonsense" like simplicial model categories works for general spaces, but if you are using any invariants like homotopy groups or singular (co)homology theories to get substantial results that do not mention those invariants, you'll probably need theorems such as Whitehead's - which means restricting to spaces homotopic to CW-complexes.)

Shape theory did go beyond spaces homotopic to CW-complexes. But being an ANR is not a shape-invariant property. It is an invariant of local shape (which Ferry, Quinn, Hughes and their collaborators do touch upon in their works) and indeed Quinn once wrote an expository paper on "Local algebraic topology". I don't think these "local" developments have ever been of interest for (mainstream) algebraic topology, but they have very good applications in geometric topology so are usually associated with the latter.

This area of geometric topology, where ANRs and topological manifolds naturally belong, has been steadily falling out of fashion with younger generations (since the 80s I would say, not 60s), apparently because it's tough enough, but not nearly as attractive for an outsider as knots, say. That might as well be a problem of the generations rather than a "flaw" in ANRs.

I think the answer has more to do with the psychology of mathematicans as a culture than with actual mathematical facts.

I was not alive during the period where ANRs were mentioned in the topology literature but I've read quite a few early topology papers and also noticed before the 60's people couldn't seem to not mention them, and afterwards they were almost never mentioned.

I think this is mostly due to the more formal side of algebraic topology, with model categories. With the terminology cofibration one could largely avoid talking about ANRs and regular neighborhoods. You of course could continue to talk about those things but if you're attempting to write something short and concise with as few confusing side-roads as possible, you would omit it.

So fairly quickly people realized they didn't need to talk about ANRs. I think this kind of thing happens fairly often in mathematics, especially when the definition of a concept maybe slightly misses the mark of what you're aiming for, or if it isn't quite as general as you really need. Terminology like this cycles in and out of mathematics fairly frequently.

You could frame this in terms of the long-term survivability of a mathematical concept -- math verbiage evolution. The flaw in ANRs is they did not anticipate that point-set foundations would become less of a focus of topology, that the field would move on and become more scaleable.