I've got a couple vehicles that you have to have the brake depressed to start them so it's a habit now. On my OBS trucks I can feel the pedal change as it starts.
I've observed the same thing on my JD7-equipped chore truck. In new condition,
the Accumulator in the Hydroboost setup stores enough assist for at least 1-2
stops after the PS pump quits turning and/or before it starts turning again. (ie: Designed
to act the same as what drivers have come to expect, based upon traditional vacuum
booster behavior once the engine has been turned off.)
In not so new condition, the Accumulator in the chore truck has lost a lot (but not all)
of it's storage capability, so I am familiar with the 'rock hard up high' brake pedal that
drops down a bit once the engine fires/PS pump starts spinning. (And this is with an
otherwise well behaved brake system that's fully adjusted out back.)
This is my first Hydroboost vehicle. Acted just differently enough that I dug around
in the FSM to find out what was going on.
EDIT: Just reread your original post. I do observe the 2 different pedal heights,
pre vs post startup, but once running my pedal is still firm/confidence inspiring.
So this *is* possible in a GMT400 vehicle? In order to figure out the 'mushy w/boost'
part, the comments by the others above may account for this. Possibly
@Schurkey
will weigh in, for he seems to have all the different GMT400 brake versions committed
to his mental Rolodex with indelible ink?
By the way, which flavor brakes do you have? That seems to be the key when these
brake questions come up.