Didn't read it all, but your interpretation of the control circuit is wrong. The control does come from the push button controller and pass through the pressure switch to the compressor relay.

What you got wrong is the action of the switch. The switch is closed by normal refrigerant pressure. Normal being just about any amount in the system up to way too much. A non running system will contain refrigerant at around 100psi (depending on temp).

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Steve Brotherton
Continental Imports
Gainesville FL
Bosch Master, ASE Master, L1
33 years MB technician