The big industrial-flavor Quincy pumps have more than one operational mode, and the unloader function is the key.
They can be run in stop-start mode... where the unloader dumps head pressure after shutdown, and keeps it in relief state until the pump reaches a fairly high speed... this keeps the motor from being loaded at a slow speed. This is normally used with electric motors, and it can also be used in engine-driven applications where the unloader's pilot pressure acts upon a little air-cylinder to idle the engine down.
They can be run in constant-speed mode... where the unloader is activated when setpoint pressure is reached... but pump keeps spinning... the pump(s) are just relieving to atmosphere... and when below the activation setpoint, the unloader releases the bypass and causes the compressor to go back to work. This is frequently used in engine-driven applications where the engine speed is not modulatable (it's powering something more important, like a synchronous generator or a traction generator, or a hydraulic power unit, or a pump).
A compressor SYSTEM may also use BOTH modes.. like an electric motor with a starting contactor, that will start and run, charging up the tank for 5 minutes, and then shutting off for a few hours... BUT... if the motor control system detects that the motor has been cycling more than so many times in an hour, it just keeps the motor running, and operates the unloader in constant-speed-mode instead... and after demand drops off, the system returns to start/stop mode.
IF the compressor system operated properly prior to arriving in your custody, then I'd bet my lunch ticket's day-stamp that one of the unloader tubes has become crimped-smashed-plugged, or broken, thus not operating properly.
Compressor valves DO get dirty... and it DOES cause them to lose sealing effectiveness, but they don't all just fail at once.
BTW, Paul- welcome to the group, and it would have been better for you to start a new thread on your Quincy 325, with that in the heading, than to tail-end onto a very old thread.