Does the mixture and throttle setting affect the quality of priming of fuel prior to start?
On a carbureted engine, the mixture and throttle setting does not affect prime, which uses a separate injection system that draws fuel into a syringe and injects it into the intake passages at the cylinders when you push in the primer's syringe plunger.  

On a fuel injected engine, where the Bendix mechanical fuel injection system is pretty common, if the fuel system is pressurized and the mixture is rich, the system is injecting sufficient fuel at each cylinder for idle continuously.  So you prime them by doing just that, pressurizing and advancing mixture, for a set period of time, say, 3 seconds or whatever, then pulling the mixture back to idle cutoff to stop spraying gas.  This does the same thing as stroking the primer on a carb engine.  Like the carb engine, it doesn't matter where the throttle is at this point.

When you start an injected engine, you start with the throttle cracked like a carb engine, but unlike a carb engine where you put the mixture to rich for the start, you have to crank with mixture at idle cutoff and move the mixture forward after the engine fires before it consumes the fuel vapour in the intake passages created by priming, within a second or two.

Injected engines are more finicky to start because the amount of prime is a bit harder to regulate, being a timing function that you have to adjust for ambient temps and engine warmup state vs counting primer strokes so they are much easier to over/under prime, and as well the timing of the advancement of the mixture after start is important.  Do it too soon and you can get flooding or intake backfires and too late it dies before it really gets going.