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Can I just get rid of the SAA box and hook an analog port up to the black/yellow and send it ring voltage to "buzz the shop"?

That depends on the "buzzer"
The SAA (Supplemental Alert Adapter) provides 48V (or -48)DC on the B\Y.
Some Alerts have dip switches that allow you to change the sound from a buzzer, to a bell or a chime.
There is also a dip switch that switches the 48V to ringing voltage on the G\R.

Ya dig? (verstehen?)