Old engine designs have larger diameter and wider bearing surfaces than modern high speed diesels. Hence more area for the oil to run out of the journals.
A little wear and the oil pressure drops especially on tickover when the oil pump is not running at optimum pressure.
I got used to not worrying too much about oil lights coming on at tickover provided they went out promptly after starting and as soon as the revs came up when hot.
Using ever thicker oil or multigrades is a bad idea, you need oil flow quickly from cold and as long as there is still any flow at all when hot the bearings will not wear unduly or seize.
The usual things should be checked of course, oil filter, oil pressure release valve, sump level, good clean oil depending on the engine.
Though not the same thing, I remember running an old Ford side valve engine with white metal bearings for months with the oil light on unless it was revving hard. When it finally grenaded and put the 2 rear con rods through the side, smashing the rear half of the camshaft off and losing all oil and water, it ran 14 miles back home on 2 cylinders, and still did not seize up.