I've studied blockchains a little bit but I don't understand how you can transfer funds between two wallets that you own without linking them. How would that be possible without either having some kind of zero sum proof (which none of the popular blockchains are using) or allowing an exploit which enables people to mint new tokens from thin air?

Generating a new public key doesn't add data to the blockchain. Only when you sign a transaction. And if you're talking identities, as in moving around websites, there's no way to link your various public addresses (logins) unless you explicitly do so using the private key you used to generate the public address. Or, you could use the same public address. So, it depends on what you want to do. Do you want privacy? Or, do you want the site to know your activity history?

The beautiful thing is, you get to choose.

You send the coins to your other wallet, exactly as if you were sending them to me. Of course you pay a fee to the miners.

Wouldn't you be able to trace that transaction back to the original wallet?

Tornado.cash is one such project. They are working on making Ethereum behave like cash in terms of privacy/anonymity.

My understanding was that using such services basically blacklists the wallet from transacting with anywhere “reputable”?

Cash isn't traced at every step into an immutable ledger. Why should cryptocurrency?

Relevant ZK podcast episode: https://zeroknowledge.fm/111-2/

I don’t see a single incorrect statement in that post.