> I've wondered if IRV couldn't be substantially improved by using ... rather than eliminating the candidate with the least first-choice options.
This is exactly the family of RCV systems -- IRV merely defines one implementation by which a winner is selected from a ranked ballot.
Borda count has its own pathologies, too -- the usual cited problem is that it fails independence of irrelevant alternatives.
https://en.wikipedia.org/wiki/Independence_of_irrelevant_alt...
(Further, Borda count fails the majority criterion, namely there are situations where a candidate is the first pick of a majority of the electorate, yet fails to win. Some proponents of Borda count argue this is a feature, not a bug.)
The wiki article for Borda count has a solid table of various voting systems and whether they satisfy various criteria:
https://en.wikipedia.org/wiki/Nanson%27s_method
and the related Baldwin's method, which, while being Condorcet methods, both fail monotonicity.
It's also sometimes called alternative vote, and the Australians call it preferential voting.
I've wondered if IRV couldn't be substantially improved by using Borda count for the elimination step rather than eliminating the candidate with the least first-choice options. (I think there's an actual name for that system, but I don't remember what it is.)