I had a car once that had that problem Would not build up enough oil pressure when cranking.. had to wire in an override button. Relay was good (Unless there was a 2nd one that was failing) and once started it ran .. Sort of but it was OLD and on it's last (over 200,000 miles on a 1992 engine is not good)
Current car is of similar age but later vintage and the newer engines last longer. So far it runs great only issue is yet another pressure switch circuit. Switch working fine. wires bad.. Just got the last bad wire dealt with Christmas EVE (Yippie) so on Christmas day 100% normal function (yippie #2) very happy.