Thank you all.

I located a very-gently-used replacement chainring for $25 and, after installing it, everything is again sweetness and light. No more slipping. Comparing the almost-new and the old, I see that some teeth have had about 1/16 of an inch shaved off. It's not a dramatic shark's fin but there is clearly a fair bit of missing metal.

So I can answer my own question. Yes, it is possible to wear out a front chainring resulting in slippage and, in future, I'll probably be replacing it when I replace the cassette.