The NTSC PSOne should sync without any problem, watch the red led, if it’s lit then the OSSC isn’t syncing to the console.
It might be that the PSOne AV connector is slightly corroded and has some unwanted resistance which might affect the sync.
Go into the Sync.opt menu and try different LPF settings (off), if that doesn’t help try raising the “Analog sync Vth” setting.
Sometime too much LPF may worsen the detection of the sync when the Vth is set lower than the sync signal.
A proper strong sync should drop to near 0v while a weak or noisy sync will remain above 0v or closer to the 300mV mark in the worst cases.
Another overlooked thing that can cause interference of sync is ground noise or buzz from the power supply, which as a result will never have a flat-line (quiet) references for the sync but spiky noise.
So going too low or high with Vth may pick up some power supply noise, LPF helps with that.
A noisy sync may look something like that (over exaggeration):
* You want Vth to be in the green (default 123mV), between signal and noise floor (reds).
Yellow = Noisy
Cyan = Clean