Yes and no.
It is obviously bad to run an engine "dry," but as long as you change your oil after running the engine, there will be enough residual oil all over the inside of the engine to keep things ok until the pump can get the new, fresh oil into the whole system. I would make certain I put in enough to deal with the drop in crankcase volume as all the lines and coolers get filled, and pre-filling the cooler may be a good idea too, but only when you drain it. Not when you are just doing a routine oil change.
As long as the engine is run regularly, and the oil is changed regularly, this should not be a problem.
Those little kart engines you saw this weekend don't even have an oil pump. They have a little dipper that sticks down off the bottom of the rod that whips up every last drop of oil into a fog. There are major bubbles in there and they turn those engines just as hard as we do.
It is about getting the oil where it needs to be, and that has to do with the oil's viscosity or thickness. Imagine this. If you poured honey through a screen, it would be slow to pass through, but would leave a good coat stuck to the screen. Now pour water through the same screen. You can flow a lot more volume in the same time, but it doesn't leave a good coat on the screen. The right oil is a compromise between the two. Thin enough to flow through the tight places in enough volume to keep things cool, but thich enough to keep metal parts from touching each other.