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).
Best Answer
This can happen, just replace the GFCI outlets in question
The TEST button on a GFCI creates an actual ground fault (about 6-8mA, maybe up to 10mA, from load-hot to line-neutral or vice versa), so if a GFCI fails to trip when the TEST button is pressed, then it's dead and needs to be replaced. This can happen due to improper wiring (backfeeding a GFCI's LOAD terminals will burn out the trip solenoid), or more commonly due to damage to the GFCI's (largely analog!!) electronics package caused by mains surges and spikes. (More modern GFCI designs may have a varistor connected across line-hot and line-neutral to help protect the electronics from surges, but this is by no means guaranteed.)