How would I prove that I own X amount of Bitcoin to somebody without violating my privacy or sending/risking any coins?
The way that this is done is to sign messages using the private keys which correspond to your UTXOs. Your UTXOs are what make up your Bitcoin balance, and by signing a message with the keys associated with them, you are proving that you have the ability to spend those UTXOs.
To verify, someone can verify the signed message, compute the addresses for the keys you used, and then lookup the addresses and see how much Bitcoin are associated with all of those addresses.
For the average person, there is software that can verify the message for you and they will return what address was used in signing that message. Then they can just take this address and look it up on a block explorer.