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 substantinal 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 natuarally 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.

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.