The problem is that in practice there are many times that the visible different between "done" and "abandoned" is null, specially for smaller single-purpose libraries.

Or mature, stable ecosystems. People use Common Lisp libraries that haven't been updated in 20 years, because they work.

I’d disagree with point 2. There shouldn’t be any responsibility for a (ex-)maintainer to provide alternatives themselves; especially when there may not be any at point of abandonment .

Is there significant value to maintaining a set of glorified search engine results in a README? It means the list of alternatives then have to be maintained in case they too are abandoned.

I find the GitHub "archive repo" functionality a great, easy way to indicate a repo is abandoned; the most you need/have to do is a little readme edit/repo description edit if you want to explain or point to a good alternative.

Problem is there's no auto detection that a repo is no longer maintained. Easy to do but github does not do it.

"No longer maintained" is hard to define, does no commits in the last year mean "this project is mature and needs no changes", "this project is dead and should not be used" or "the author still cares about this project but is busy with other stuff this year" or something else?

Exactly. It is subjective. One just needs to look at the contribution history and determine if it is what they need.

Depends on what the repo is/problem it solves. Sometimes something is considered "done", when it doesn't need anymore work to it. :-)

How do you define if a small library has been stable for years?

Issues filed. Their recency and concerns.