Reely,
Habit. Universal procedure. You give a bit of fuel pressure to deal with once you move mixture to RICH in case there is some vapor lurking before the servo. Watch the gauge next time you do it.
Take a look at the fuel lines with respect to residual heat.
The reason they start when warm is the residual fuel that must bleed out of the lines into the inlet when hot after shutdown is a theory.
Lots of folks do it this way. It works.
Your S 300 if properly maintained should start just like your car. If it doesn't there is something wrong with it.