I am curious as to the exact process on how to sign/ prove ownership of a BTC address with how I have seen many users do it here in this section:
Think about it, every man who self-proclaimed that they are Satoshi Nakamoto can only be prove by signing a message with those address known to be owned by Satoshi, no ifs and no buts, if they can't then they are just want an attention. It's the same way how that thread works, but since there are already recovery team, it make so much easier to recover an account now than before.
Would signing that message in any way compomise the security of the private key? What are the drawbacks to doing this for ones security both of their keys and of their location?
Your question to chatgpt is out of this place, you only compromise your private keys if someone knows other than you, so you need to save it to a secured and safe place. There's also no drawbacks in signing a message, location detection is something like having a raw data of your latitude and longitude which is no way to be found on your keys.