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.
