So how do you do that without creating a link to your real identity? That’s the part I find confusing. It’s not about Bitcoin itself, it’s about the process of converting the fiat to Bitcoin in a way that doesn’t leave a trace.
Simple way is to exchange cash for bitcoin with someone using p2p trading, this won't leave any trace.
For this you need to have someone in your local area who is interested in trading, and you can get connected in different ways.
Other way is by using decentralized exchanges and services like PeachBitcoin, Bisq, Robosats, etc.