How important is 1psi of fuel pressure?

Except when it doesn't.

It depends on the type of mechanical pump, and also depends on the type of regulator if one is used.

Oh wait. I already posted that. Oh well. redundent post.
Easy way to show this is, take a garden hose, no nozzle, just free flowing. Then take same hose, put a nozzle on it. Which flows more fluid and which has higher back pressure...