Many PSU after about 2005 are too new ATX spec to be turned on by an older board unless they state ATX 2.03 compatible iirc.... there is a good topic here about new ATX PSU on older ATX boards but cannot find it right now....
You can try loading the +12v line with a few hard drives (do not have to be connected to IDE, just power) and see if that helps. If not you could temp jump the Power On line (green) to Ground (black) to see if that powers it up...

"modern" ATX PSU means almost nothing these days. Reason why is because there are still "modern" no-name garbage and/or cheaper PSUs that are really an old 80's half-bridge group-regulated design or 90's STF group-regulated design (same as many PSUs from the 90's and early 2000's), but just "adjusted" to supply more of their power on the 12V rail than then 5V rail. These PSUs, when connected to an old motherboard that draws most of its power from the 5V rail rather than the 12V rail like "modern" mobos (or really, just any motherboard that doesn't have a 4-pin or 8-pin 12V EPS connector), will tend to get a saggy 5v rail and really high 12V rail - sometimes enough to trip the PSU's protections. Not all modern PSUs will act like this, though. In fact, just about everything that's not a complete no-name brand or not too cheap will use DC-DC modular boards to generate the 5V and 3.3V rails. PSUs with those will *usually* not have an issue to power older 5V-heavy(ish) motherboards, so long as the load on the 5V rail doesn't go beyond what the PSU can supply (e.g. a socket 462 motherboard with a top-end Athlon XP that requires about 70W of power... which is about 15 Amps from the 5V rail for that alone and not counting the chipsets or anything else.)

So depending on what you have for your "modern" PSU, that may or may not be the issue here. Please post what brand and model you have, so we can try to figure out if that's a unit that may not like cross-loading.

Worth nothing is that some older (but still pretty modern, at least by the board's standards) higher power PSUs (typically over 500-600W) might not run correctly or at all with low loads... and your PII board could qualify here (as PII's are generally in the 30-40 Watts TDP range - and almost all of that from the 5V and 3.3V rails.)

That said, if you were able to make the board POST several times with your modern PSU and it ran stable with it, then I suspect the PSU may not be the issue here and you might just be dealing with an intermittent problem or a general problem elsewhere (e.g. electrolytic caps on the motherboard).

In any case, while at it, also post the brand and model number of the old PSU. Some old PSUs pretty decent and worth refurbishing to use with older systems like yours. Most of the time, they just need a recap. If you've soldered before, then that can be a relatively easy job.