Knowing how shut trips (ST) are wired as well as how they function to trip a breaker you question certainly doesn't make sense.
I would question the 208v ST coil and why it would be used since a 208v 3ph, 4w system does have 120v L-N voltage available which is much safer that using 208v L-L.
The next thing that I would question is how the ST is powered. If in fact it truly 208v the breaker that feed is must be a 2p breaker and, being a L-L voltage, that the switch that activated the ST must be 2p.
And the chattering coil doesn't make since a 120v ST should still activate immediately with 208v applied to it even if the coil isn't rated for 208v. However, since STs active instantaneous when their rated voltage is applied, it hits the trip bar of the breaker, tripping the breaker. A cut-off switch is wired in series with the ST coil immediately opens and removes power to the coil even though power is still being applied to the TS circuit. Because the ST activates so quickly and the circuit cleared by the cut-off switch and overvoltage really isn't maintained for a long enough period if time as to damage the coil.
The only time I have seen a ST coil buzz or chatter is if a marginally undervoltage is applied to it which is not quite enough for the plunger of the solenoid to be activated which can cause buzzing. When a ST doesn't have enough voltage applied to it in order for it to active an open it's cut-off switch you have a situation where the voltage that is applied is now not able to be disconnected. This allows current to continue to flow through the ST coil and will burn it out in short order.