Effect of wire gauge change on thermocouple measurement
https://solutions.borderstates.com/what-are-the-different-types-of-thermocouple-wire/

https://www.omega.ca/en/resources/thermocouple-wire

These websites state that the reason for a larger AWG is to reduce the wire resistance in longer runs. The resistance needs to be kept sufficiently low so that noise currents induced onto the wires do not produce a noise voltage that drowns out the thermocouple junction voltage from the Seebeck effect.

The longer the run or the more EMI in the area, the larger the AWG has to be.

Rule of thumb is to try to have a loop resistance of less than 100 Ohms. 20AWG works fine up to 100 feet in an area with no EMI.

12AWG would be expensive. That's a lot of nickel and chromium. Your thermal response would also be very slow due to high mass.

Note that you can filter thermocouples with resistors and capacitor. A series resistor on each line, and on the far side of the resistor away from the junction put a capacitor between the lines and a capacitor from each line to ground. Use C0G capacitors if possible since these are in the signal path (no piezo effects or DC bias effects). All placed as close as possible to the input, of course.