BTC never requires any personal information.
People are often confused about this, considering some websites and services ask for KYC.
These are just businesses who market themselves as essential in the process.
This is not the case.
You can create BTC paper wallet on your own without ever visiting any site.
You do however need internet to broadcast a BTC transaction - message signed with your private key.
But you certainly do not need popular commercial websites to give you permission to do so: coinbase, binance or any similar.
Proxy is not what protects your BTC anonymity. Nobody cares where this transaction broadcast comes from.
It's how the BTC is used that matters, from whom you receive coins, and who you send it to, paired with other data, such as communication channels, exchange of goods timestamps. It mostly depends on culpability of actors you trade with.