I've had lots of high mileage cars before but never had a pump go bad. Had a clogged pickup once which was confusing: Good pressure until I had a few minutes at high revs then the oil pressure would recover once I dropped back to idle.
I don't understand how the pressure ever comes up at startup. If the pump is loose enough that it doesn't immediately pressurize, what is changing in the next 10 seconds? I'm sure you're not reving it up (I certainly wouldn't if it's making noises).
Maybe the solution is "never turn the engine off". Hope this helps.