Overlays are a genuinely contested topic in the accessibility field. Independent testing and numerous documented lawsuits have found that overlays frequently fail to fix, and sometimes actively worsen, real barriers for screen reader and keyboard users, because most accessibility problems (broken semantics, missing form labels, keyboard traps) live in the underlying code and can’t be reliably patched by a script layered on top.
Overlays also do not provide legal protection from ADA lawsuits; several companies using them have still been sued. Genuine WCAG conformance, not an overlay, is the standard both courts and disabled users actually rely on.