The function of the Booster is to make it easier to push the brake pedal. Try stopping your car (not this one) when the engine is shut off - it will be hard to push the pedal. I'm not controdicting your mechanic on the Booster being bad - I have heard of this, but its rare. Here's what you should check, in order. Its long, but spongy pedal can be caused by lots of things (possibly even the booster).
Front brakes in good shape - pads are thick, rotors are smooth
Front brakes correctly assembled
Front brake calipers not hanging up on bracket
Unbolt front calipers and check for smooth piston movement
Rear brakes in good shape - pads are thick, drums are smooth
Rear brakes correctly assembled
Rear brakes are clean and correctly adjusted - should be slight drag (slight)
No leaks in system anywhere - No evidence of fluid leaking
Check all 4 wheels, Master Cylinder, lines, Proportion valve, under dash at rod connection
Check hoses for cracking or damage - replace
System has no air in it - Bleed and pump new fluid through
Pressure bleed brakes with pressure unit
Possible Master Cylinder leaking internally
MC will usually leak at the back, unbolt the two achor bolts and look
Hope this helps