Firstly, there's no such thing as a fast stability test, everything has to heatsoak up to maximum operating temperatures. If you just want something that lies to make you feel good, pasting a line like this
PRINT "Congratulations your system is 110% stable, the stablest stable of all stables, you are also very clever and handsome."
Into your favored compiler, qbasic, turbo whatever, and compiling it to an executable gives you a program that is 99% as accurate as anything that tells you your system is stable in less than 2 hours.
Secondly, the best stability test for everything from 386 until PII is Doom, period.
It is also "an opinion" to consult for PII to P4 class, but it is not 100% on faster systems as they have areas to be tweaky that Doom's code gets nowhere near. Quake is not a good stability test. I mean it is if all you want to do is play Quake, but you can get systems set up to play Quake at highest frame rates and refuse to boot windows and crash in many other games. Obviously if you can't even get the system to run Quake without crashing, it's not stable in the slightest.
So stability testing with Doom, first, run it as the normal benchmark, though not Phil method, edge to edge screen and HUD bar, high res. If it crashes out before completion, super unstable. If it completes but scores (thandor.net etc) are not "close" to similar systems but like twice as fast, it short circuited the demo, watch it next time, it probably goes out of synch. Also super unstable... completes with a score in the ballpark of expected. Good, first hurdle, not that unstable. Next, put the demo on loop for 2 hours, this is the heatsoak test, if it doesn't crash or go out of sync in 2 hours, you can now call it fairly stable. This might be good enough for you.
If you still have weird win 3.11 errors and odd crashes in other games, you might want to leave demo looping overnight, if it does 24 hours, no synch loss, running perfect at the end, you can be fairly sure it's not hardware related as far as the subset Doom uses goes, it could be related to conflicts with other cards and features unused by Doom still but your integer/cache/lower 8MB are all behaving. In case of windows errors at this point, you can probably start to investigate software and other causes, like known bugs.
Unicorn herding operations are proceeding, but all the totes of hens teeth and barrels of rocking horse poop give them plenty of hiding spots.