Why would a "power switch" be designed this way?

It looks like the purpose of the circuit is to power-down at current zero-crossing (which would alleviate voltage spikes from interrupt current through an inductance, if there are any).

When triggered, SCRs latch on and thereafter continue to conduct current. They only enter the blocking state if the trigger signal is removed and something else in the circuit reduces the current to zero through the SCR. That means SCRs can only turn off at a current zero-cross which happens naturally for an AC source. For a DC source it would never turn off unless another component in the circuit interrupted the current.

That circuit would allow the bus analyzer to be turned off very gently.

I can't answer why the bus analyzer needs that though. I don't know if something bad would happen if it didn't have that or if it's just really high quality.

EDIT: @P2000 mentioned EMI from said voltage spikes. Since this is an avionics bus analyzer and is powered off 400Hz rather than 60Hz, it makes me think that it is expected to be used in-flight. In which case such a switch would make total sense so as not to not interfere with the avionics currently running when it is switched off.

EDIT: @TooTea mentioned that this eliminates the effects of mechanical contact bouncing on startup which may result in sparks which would reduce EMI on turn on.

it's purpose is simply to allow a small switch to control a large current.

when it switch is on any current attempting to flow from A to B or the other way will trigger one of the SCRs

when it is off it will block AC.