You're already asking a lot of users. Let them save the recovery codes wherever they want! The real point of the codes is to prevent people abandoning 2FA in frustration, even if it is at the expense of some security.
I agree it's a lot to ask. But I think it's better to demand extreme security considerations from users and fall slightly short in practice than to encourage any practice that improves account security beyond a single password (e.g., SMS codes).
One frustrates users and makes them abandon 2FA, the other (IMO) encourages some complacency and makes it harder to justify changes to users (especially ones that are really great from both UX and security perspectives, like WebAuthn).
Do I understand you right as arguing that having some people not use 2FA because the requirements are too harsh is better than bringing people onto "2FA lite"? Or are you saying something different?
I don't think the logic you're employing here is coherent. Consider it a bit longer! This post could be a good longstanding reference, but this is a big flaw.
Edit: Updated.
I have multiple U2F keys configured on all of my important accounts. I'm comfortable enough in my belief that I won't lose _all_ of my keys to not want recovery codes to exist so I don't need to worry about storing them. This places me in the extreme minority of users to be certain, but I still don't want my security weakened by recovery codes that I won't ever use.