I think a good starting answer is
(1) - It would stop the case from becoming live if something shorted to the case would it not? :-)
(2) BUT, not only stop it becoming live, but also provide the intended means of operating the provided protection equipment if a phase to case fault occurred.
aka "blowing the fuse" or "tripping the breaker" as the case may be.
The operation of fuse or breaker in the case of a fault is probably more important than keeping the case from reaching mains voltage at all. Both achieve much the same result in terms of case potential above ground BUT breaker/fuse tripping both tells you there is something wrong and removes a potentially lethal problem which may otherwise manifest in some different manner.
(3) Along the way it may stop nuisance shocks which occur from current through any X and Y caps that may be fitted (probably none in your power supply).
Also ground referencing your supply stops the whole assembly wandering off to a semi-random voltage of it's choice relative to ground that has nothing to do with the power supply proper. eg in some cases an ungrounded case may be driven by electrostatic charge from the effect of carpet on clothes etc and gain a voltage of thousands of volts (very literally) relative to ground. Touc it when grounded and you may feel a small or not so small kick as the stored energy "discharges". What feels small to you may be the last thing that your circuitry ever "feels".
Not a real answer, but it got too long for a comment.
There is not really enough information to give a really good answer. It may work OK with everything grounded to the case. EMC regulations are performance based. The FCC doesn't tell you how to ground things. It just tells you how much RF you are allowed to radiate into space. If you have a bunch of stuff inside a metal chassis, and grounded to it, you have a good chance of passing radiated emissions just because your metal chassis will act as a Faraday cage.
But, the most important thing is that you MUST run a GND wire from board to board with every serial connection. So if the main board has an SPI interface connected to a peripheral mini-board, you must run a dedicated GND wire along with the other SPI wires. Preferably, the GND should be twisted with the SPI clock wire. If possible, it would be even better to run a GND wire along with every signal, and twist it with its signal wire. It is almost like you are wiring a differential pair, with the signal being one element of the pair, and the GND being the other element of the pair. This will greatly improve (reduce) radiated emissions, and also preserve signal integrity.
It doesn't sound like there is any major safety concern (as far as shock hazard goes). You just need to give some consideration to wires and fuses and short-circuits. You just don't want a wire to ignite a fire if it carries a large fault current as a result of a short-circuit somewhere. A few strategic fuses can probably take care of this.
The only shock hazard would be associated with how the AC power enters the box and how the chassis is grounded to the protective earth ground. I am not an expert on that stuff. I think you just need one solid, low-resistance connection from protective earth to the chassis. Sometimes a chassis will have a metal stud for this purpose.
Best Answer
If the circuit is already isolated then connect the varistors to the local electronic ground. The varistors should be rated 50% higher voltage than the highest signal voltage. If grounding to earth was needed for your AC power, that would normally be done at your circuit-breaker panel.