Assume we power up the shunt trip from phase A-B and that the shunt trip is used to trip the breaker when a fault occurs. When a phase-phase fault occurs on A-B, the A-B voltage drops to zero and there is no power left to operate the shunt trip and clear the fault.
Using DC power reduces the chance of control power not being available for tripping during a problem.
If the shunt trip is only to turn off vent fans in a fire, this may not be an issue.
Many 480/277 ground fault protection systems use AC power tapped from the line side of the breaker or switch for the shunt trip. Power is taken phase-phase (480V). The shunt trip, the relay and the associated control power transformers are designed to operate reliably on 277V input to the 480V power, as will happen during a ground fault on one of the two phases. A good test on a new GF installationis to power the GF relay circuit at 277 and verify that it will trip.