My mower is very simple without a fuel pump, so a little different. Fuel flows by gravity into carburetor.
As I understand it, what makes generator engines a little different than newer cars is the location of the fuel pump. On newer cars and trucks the fuel pump is inside the fuel tank, so it pushes the fuel to the engine. That’s very effective until you have to replace the fuel pump. Just had to do it.
My generator had fuel pump at the generator, so it had to suck the fuel from the vehicle’s fuel tank. This is the way old cars like my Mustang are set up. Old cars normally had a mechanical fuel pump at engine, and if you didn’t run them for a while, sometimes required cranking to fill the entire fuel system. With electric fuel pumps that operate independent of engine cranking, it can save battery capacity by just running fuel pump.
I suppose that proximity of generator to vehicle fuel tank and also relative elevations “may” affect priming some, but I’m not sure of that either. If so, not all generators will prime as quickly. Haven’t looked into this much because my Onan, when I had one, usually started quickly.