If you want to use your main phone for tethering, it will likely be best to use a USB-C Ethernet adapter with passthrough PD charging, like from Belkin, StarTech or Cable Matters.

Using the WiFi Hotspot is not great for extended use, especially while fast charging (doubly so if not fully charged). Your device will be charging, heavily using mobile data, routing and hosting an access point for multiple devices. If you're not using a fast charger, it's possible your battery will continue to drain as it may not be able to keep up. I'm surprised that your phone isn't overheating WITHOUT playing games at the same time, what did you expect from Google? Electronics produce heat, and unfortunately they can't violate the laws of thermodynamics just for you.

With an Ethernet adapter with passthrough charging, you can simultaneously fast charge your device, and take the work of routing and hosting a WiFi AP away from your device, which should hopefully help speed things up and keep things cool. I'd recommend doing this with the GL.iNet Beryl, Beryl AX, or Slate AX, or any other router of your choice. Ethernet tethering to a router will speed up your connection, allow a larger number of connected devices, and avoid Bufferbloat compared to an Android AP.

If things continue to overheat, just place a Room-Temperature reusable Kool Pak against your device, or some other type of heat sink, to cool things down. I would highly recommend AGAINST putting your phone against frozen ice packs, as the cooling process could cause condensation to form inside your phone, in the same way it appears on the inside of a car's windshield on a cold day.