There is an easy test to check for air in the brakes.
With the emergency brake applied to move the pads or shoes into contact with the disc/drum.
With the Master cylinder cap on, have the brake pedal pumped a coupla times and hold down.
Remove the cap off the master cylinder, watch the fluid surface and have the pedal released.
Any air in the system is being compressed by the pedal, - when released, the compressed air will force fluid back into the reservoir.
There should be very little return to the master, cuz, with the emergency on, - the springs can't compress the wheel cyls, - the pads (front and rear) only "relax" displacing nothing,
It would be expected to see a slight "roiling" of brake fluid in the reservoir, but any volume of return indicates air.
I have had a literal geyser of fluid shoot violently back after customer swore he bled brakes.
Brake fluid is an excellent paint remover, so if any fluid gets on paint, have lotsa water nearby to flush liberally .
No volume of fluid return, - no air,
Experience .