That works, to varying degrees, until the provider gets a letter from law enforcement. And unless you find some provider willing to take crypto, you are doxxing yourself when you pay.
Several domain registrars / hosting providers (including Njalla) accept Monero, which is as untraceable as you can be. Or Monero can be used to indirectly pay a bitcoin invoice very easily (even Namecheap accepts bitcoin)
It doesn't matter. You can buy Bitcoin with cash, convert the Bitcoin to Monero and then as soon as you send that Monero to your own wallet, you're done. You have digital cash to spend anonymously.