How important is a good rack?

There's racks and there's racks with vibration control. One gets your equipment off the floor, the other does that while reducing distortion. Some of the latter are more effective at distortion reduction than others. Devices that are highly effective at vibration control and look great, such as those from Silent Running Audio or Harmonic Resolution Systems, command top dollar. Audio room infrastructure (acoustics, power, vibration control) is a necessary condition for realizing the sonic value of electronics and speakers.
 

Racks are a clear case of "horses-for-courses". Some manufacturers e.g. Grand Prix Audio, will insist on knowing what the various loads on the rack will be so that they can customise it by tuning it.
Praiseworthy though this approach may be it only deals with synergy at a simplistic level (i.e. no real-world AB testing with operational equipment)

Turntable support synergy is so unpredictable that the simplest mechanical features can radically alter the sound. I doubt that there are any guarantees that an esoteric table costing megabucks would even top a group table in a 5 or 6 way face-off against bargain priced opposition.

How does an equipment rack remove distortion?

I know what you mean by not wishing to put the things on the floor. Everyone is like that. But when you go to the shows many of the best sounding rooms have the equipment on the, uh, you guessed it. The floor. It's easier to control what's going on.

10 yrs ago I was using a generic Sound Organization rack and came across a used Billy Bags rack. The only reason I pulled the trigger was the price ($600. The rack cost twice that new!) and I liked the look of the rack. I wasn't considering sonics at all, I didn't think it would make much of a difference. I was wrong. I immediately heard an overall improvement, everything was the same as before except for the rack so....not sure if Billy Bags is still in business but if you can find one 2nd hand I highly recommend them. As far as my TT's concerned, I ordered my Linn LP12 and bought an Italian custom made stand (low mass, high rigidity, low to the ground, spikes on top and bottom). This was 25 yrs ago and haven't felt any need or urges to change anything!