Here's what I found quickly.
Someone also mentioned the rim can be stuck to the mount. Try kicking it.
"Try and get as much WD-40 on the carrier as you can. Let it soak in and spray again. Have someone under the spare pushing/pulling it as you try to turn. Once I broke mine free, the steel rope frayed, and this may have been stopping it from lowering. Still using it and spraying it often just to keep it free. While you have the WD-40, shoot up the tail gate key lock as this gets really dirty and also freezes up."
"I had the same issue. My TPMS triggered a low air warning and it was in the spare. The oil change place told me that the tire valve cap was frozen on the TPMS sensor and they did not want to try and remove it. They suggested that I go to Discount Tire to try and get a new TPMS sensor for the tire (not that i wanted to for over 100 bucks!) So I did my tire lowering mechanism was frozen solid just like yours. So they crawled under the runner and sprayed their version of liquid wrench on the cable mechanism that holds the tire up. My recollection is that they also sprayed some in the hole of the bumper where you insert the tool. It took several squirts over 20 minutes before they could get it loose. Finally, it came free and they didn't charge me a thing."
2016 Toyota Tundra SR5, 5.7 V8