The very early Norton dommies had an oil pressure gauge fitted in the tank.
Calibrated to 160 psi, it must be said - but with an oil pressure relief valve, like you say.
So the pressure was limited to 40 psi max, except when cold - when the oils in those days was somewhat thick.
Oil gauge showed some scary low values, and Nortons answer was to delete it for the swingarm models. !
The engines ran quite well of course, and didn't seem to mind low pressures.
The oil tank doesn't get particulary hot, even on these iron engines, so the oil does little with respect to heat removal ?
Except around the exhaust valve, when every little bit helps.
It was the scavenge-fed oil supply to the head that was somewhat lacking, the oiling being somewhat intermittent.
This showed up as sometimes rapid wear of cams and cam followers, due to little oil coming down the pushrod tunnels.
And also of rockers and valve guides.
It wasn't until the pressure fed oiling to the head for 1964 (was it ?) that this was corrected.
And the real reason for the bigger oil capacity oil pump, since pressure oiling the head would otherwise have taken pressure/volume away from the big ends...