There is sadly no hdmi CEC support on Most GPUs for pcs. So when you want your tv to turn on and off with your pc it is only possible with the tv being reachable from your pc via ip.

I believe LG doesn’t advertise such an api via Bluetooth

You can get USB controlled CEC injectors for HDMI.

https://www.pulse-eight.com/p/104/usb-hdmi-cec-adapter

For a 4K120Hz TV I'd be worried about this adding signal integrity issues - "all versions of HDMI®, including HDMI® 2.0a" doesn't inspire confidence.

My understanding is that CEC can be received on any alive interface, not just the active.

Yes but you will lose 120 FPS/VRR. So for high end TVs not really ideal.

HDMI breakout and a esp32 module flashed with esphome is another option. The tv most likely doesn’t care that the port that sent on/off command is not the same port the video signal is coming from..

The tv does care which port it came from, since it automatically switches to the one which made the request.

One could build a IR emitter but you need soldering skills for that or buy a usb one which cost north of 30€ + shipping.