How important is a good rack?

Yes, suspended tables like the AR prefer a light rigid platform.

If your turntable has 3 feet then that should influence your decision. 3 feet are also intrinsically more stable and easier to level.
If you have a dense concrete floor also consider siting the T/T on a separate table or rack/shelf that includes an additional shelf for the phono stage/PSU.
If wall mounting always mount the T/T shelf on an external solid wall made from bricks, concrete etc.
Keep vibration sources such as transformers off the T/t shelf. Try to get some form of isolation between shelves e.g. spikes & cups
Watch the cable dressing at the back of the turntable i.e. don't crush, mix or tension the cables, they should be as free as possible.

A facility to level the shelf i.e. T/T platter, may be essential if the T/T has no levelling feature.
Suspended tables are awkward, you are better off using mini spirit levels in 2 diametrically opposite positions to stop the platter becoming unbalanced.

One way to test the shelf brackets is to hang from them. If they flex under your weight like a springboard they aren't rigid enough. Solid triangular brackets are better for that reason.
The shelf material itself should be light and rigid e.g. wood which is thick enough to be considered rigid (you can also screw spikes into wood for decoupling....)
Ideally that shelf should rest lightly on spikes rather than physically connected to the brackets.

Final word, after you've done it enjoy the music ;^)

FWIW, the designers of the Continuum turntable (quite expensive?) and it's integral stand said that despite the massive cost, effort and design simulation that went into creating the stand they admitted it was still a poor substitute for a good wall shelf (!)

I must be a pervert - I still use a table ;^)

I'm not into wall shelves myself but some swear by them.

I suppose the wall would make for a better foundation than many floors, especially in modern houses with suspended plywood flooring that has a lot of give. Nice during an earthquake, but not so much when the tonearm starts flying if you start to boogie down.

What you are trying to accomplish is isolating external vibrations from your components. The best rack will accomplish little if it is sitting on a bouncy floor. Wall mounting on a bearing wall usually results in the best isolation.

Best to have 'rackless' room, say hiding the stuff in a closet where the acoustic energy that the speakers produce will not impact the rack and its components. Very important if you have a turntable in your system.