Can I set up a new Apple TV to a newer LG television using the USB-C connection

No, you must connect it using an HDMI connection.

That's not the way I'd go personally but you could. If you are not using a HT Receiver then I'd connect Apple TV into of the LG's HDMI in ports and the Cable TV to another. I happen to use a HT receiver and let the receiver do all the switching for me. I also use a universal remote that I set up so it automatically changes the TV's input, controls changing channels (or apps on the ATV), volume, power, etc... This is much more elegant IMHO.