Edit: would a metal aerogel not count? (Definitely not maker ready though, I assume)
Typically the fluid will be air which sucks at moving heat (hence why aerogels are great insulators as they hold a lot of air).
I'm not positive but I suspect that an aerogel in a higher specific heat fluid wouldn't do any better as the flow volume through the aerogel would be quite low bounded by the volume of the aerogel.
If you want to cool the hotside of a TEC you need to push a lot of watts per unit area. The best things for that are high specific heat fluids which are often used in compressor systems and then they're termed refrigerants so you're back to traditional air conditioning. The other option is very cold fluids (liquid nitrogen) but at that point you may be better off bringing the LN2 into more direct contact with whatever you're trying to cool.
A heat sink needs high thermal conductivity. Hence why most heat sinks you see are metal.
A much better solution are heat pipes, which utilize convection internally to create very high effective thermal conductivity. Assuming the system is not in continual use, the need for a large radiator can be avoided, by using a tank of water or oil (if electrical conductivity is a concern) to dump heat into. With a tank of water, you can also cool below ambient temperature by using evaporative cooling.
But while most people are talking about Peltier as a cooling module, note that it can be used as a more efficient heater. You can spend 36W of power (12V at 3A), and move maybe 10W. That's a total of 46W of Heat!
In essence: a Peltier is always going to be a more efficient heating device than a standard coil. Heating would be sufficient for tasks like Plant Seed germination or Bread Proofing... seeds / bread want to be around 90F or 32C, so you'd need to create a 110F / 43C source of heat.
Classically, people used light-bulbs to heat dough or seeds for germination, but a Peltier would be easier to control and more efficient.
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Refrigeration is cool (hah, punny), but precise heating also has a lot of good uses... even at the only ~20C or 36F differential where Peltiers are practical. Going +/- 20C or 36F is still a useful trick.