As I have one of these B24’s, I thought I’d weigh in.
The thing I think the bladder does is a sort of capacitor for air – a capacitor or ‘tank’ in a circuit can be used to supply surges of current that otherwise would incur losses (proportional to its current) as it travels through wires and connections.
So what the bladders does is like a capacitor increases the supply of charge to rush into an inductor, increases the supply of air immediately available to rush into the opening, sounding the reed.
I have a tendency to stop blowing between notes on my alto/soprano melodicas as a form of expression while playing them – it alters the attack form in some way that my brain likes to play with. Or maybe its left over from trumpet playing. Anyway. When I play the B24 with that pressure dropping technique the bladder isn’t charged and the notes (especially the low ones) on the B24 won’t sound! Every time I pick the thing up I have to remind myself to maintain a steady pressure rather than relaxing it. And even then its a touch tricky.