Preface to OP: I am a dope. The reason the GFCI does not trip when I push the "test" button is because the GFCI has failed. That's the whole point of the "test" button. End of story. Read on only if you want more details about how the things work.
I've noticed a GFCI outlet in which the "test" button depresses but does not cause a trip. Pressing the "reset" button temporarily disables the outlet, until it is released (which may be normal behavior).
I don't have a GFCI tester, so I tested it by shorting a 10K resistor between hot & ground. That's 12ma or so, which appears to be well over a typical trip threshold; and I've tried it on newer ones (elsewhere in the house) and it trips them just fine.
Furthermore, I have TWO GFCI outlets that are exhibiting this exact same behavior. However, they are the same brand and were installed when the house was built 32 years ago. Is this a known failure mode ? Seems like the problems most everyone reports are needless trips.
P.S. They were not wired incorrectly (e.g. "hot" to "load" terminals).