I know a fair bit about cars...
... and some of it might just translate.
If you think of the combination of tyre and suspension as the total suspension system, its job is to be a low pass filter. The buzz and bumps of the terrain should be evened out, in order for the "signal" of terrain changes to come through loud and clear to the rider and for rider inputs to be played back to the contact patches in the same way, whether they be steering, acceleration or braking.
The essence of the job of the tyre/suspension is this frequency dependent filtration. The tyre's job is to take out the higher frequencies (of small amplitude); the suspension takes out the slightly slower frequencies which tend to include bigger amplitudes. The grip you experience is partly a function of the contact surfaces (tyre tread appropriate to the ground conditions, with a contact patch shape determined by pressure and load) but is a large part influenced by the frequency response of the tyre and suspension (legs/arms too with a skilled rider out of the saddle). You want the suspension and tyre to work in partnership with a gradual handover of response as input frequency varies.
This is why running a big balloon tyre on a narrow rim with a racy, taut suspension setup is a nonsense; the suspension will fight the tyre to aborv the same range of frequencies.
This is why a 1.85in Smoke Lite Comp run at 55 psi with a 6 inch travel all mountain rig will not work right; there is no chance that the performance of the tyre will neatly handover to the suspension, leaving you with a bunch of frequencies that are unhandled (grip will probably be poor).
Because MTBers are independent-minded types and everyone seeks out their own sweet spot, you do get wildly varying advice on tyres/rims/pressures/suspension. I try to stick to the principle of keeping all components in proportion. If I am setting up a racy bike, the suspension will be firmer and have less travel; the tyres will be smaller; they will sit comfortably on a narrower rim. If I am setting up an all mountain bike, the tyre suspension and rim will all be proportionately larger and the speed of response of the ride will be lower.
All of this is of course complicated by the other significant factor in suspension calculations... the sprung mass. Rider mass over each axle varies hugely just through body position changes, without even considering variations between riders. If you have a hardtail, one end of the bike is going to be chasing radically different frequency responses to the other end. This all represents fantastic, challenging arena in which to constantly play with setup, with self-delusion telling you whether or not you've got the answer right.
