Clearly you don't understand the construction of your engine, If you pump the primer and then leave the aircraft all the fuel will run down into the airbox.
The practice of priming the engine and then doing the walk round inspection is just a good way of setting up your self for an intake fire.
That really depends on the position of the valves at the time, and which cylinders have been selected for the prime. It also depends on the engine, and the induction arrangement. Fuel may end up in the induction, exhaust, cylinder, or pooled in the cylinder head by the intake valves.
Bear in mind that with a cold engine during a cold weather start in the big radials, we filled the supercharger with avgas while cranking. I knew an old captain who used to say that unless you could get at least a five gallon bucket draining out the supercharger drain during the start, you weren't flooding it enough to get a reliable start. In a light airplane engine, starting off the mixture and adding fuel as you crank is probably a much better idea. Just rememeber lean when hot, rich when cold...and forget about the accelerator pump. Leave the throttle alone during the start.