A simple explanation: Temperature control, is certain alloys change resistance on a consistent manner. Since many metals change resistance at predictable manner, it can be used to tell the temperature of the coil in a fairly accurate manner based on the change of resistance from room temperature. The Coefficient rating is the difference in resistance based on heat = X temperature. This is why when you get a new temperature control wire, you must determine it's base resistance at room temperature. Since each version of stainless is a different mix of metals, not all metals are listed. Other settings are known, like solid titanium or solid nickel are not a mixture of metals, so the coefficient is already known. For Stainless Steel, there are different mixes of Stainless Steel, but since most vaping wire uses Stainless Steel 316L, it is assuming that SS is using 316L.
When a mod uses temperature control, it provides a heafty load of power until the resistance reaches what should be whatever resistance that wire would be at the temperature you set at. If you use Nichrome or Kanthal, those don't change resistance as much as Stainless, Titanium, or Nickel, so it would simply keep providing a heafty load... constantly. Since the wire would never reach the resistance change, it would just burn your coil.
TCR setting is if you wanted to use a wire other than 316L, Nickel, or Titanium.. or a wire that the mod doesn't support. If you know it's Coefficient rating, you can plug that rating in and use it in temperature control. Here's a list of some Coefficient ratings I found.
Ni200: 0.00620 (Nickel)
Ti01: 0.00350 (Titanium)
SS316L: 0.00092 (Most Common SS wire)
SS316: 0.000915
SS317L: 0.00088
SS317: 0.000875
SS304: 0.00105
SS430: 0.00138 (shouldn't be dry burned very very hot as it becomes brittle at very high temperatures but a standard dry burn is no problem)
SS410: 0.00155
NiFe30: 0.00320
Tungsten: 0.00450
Platinum: 0.00392
Gold: 0.00340
Iron: 0.00500
The list did have Nichrome on it, I removed it from the list, since there are different versions of it. There is Nichrome 80, 60, and 40.