Thermal pads, their thickness and conductivity
Thermal pads work basically on the basis of the fact that they're more conductive then the material they replace (air).
They don't work too well because they generally need be soft enough to conform to the surface they're trying to mate, which places limits on their composition.
Basically, you want the thinnest thermal pad you can manage. The obvious end-point for this is thermal paste (where the thermal paste can squeeze out allowing the thinnest bond-line possible).
Generally, you design for the thinnest gap possible. You only go thicker due to mechanical constraints.
Thermal pads are basically a compromise that you only take due to mechanical constraints. They're generally never better then a mechanically clamped interface with thermal paste. A thermal pad provides assembly convenience (and substantial gap filling in some cases), but poorer performance.

more specifically why does their thickness matter so much even more so than their actual conductivity numbers

This is pretty apparent if you consider it a bit. If you reduce the thickness of a thermal pad by 1/2, you've effectively cut it's thermal impedance by half, which would require double the thermal conductivity to equal. It's much easier to thin the thermal pad then to improve the thermal conductivity beyond material limits.

Both that and thermal pads for example on top of PCB modules, VRAMs, VRMs etc, but ultimately thermal pads that dissipatiate heat.

In any application I've seen, thermal pads do not dissipate heat. They move heat from a device to a dissipative radiator.
Making a heatsink out of thermal pad material would be silly, because they're really quite crappy in terms of actual thermal conductivity.