It's not a fake e-mail, it's a legitimate e-mail that you can send e-mail to and the user will receive, which has to be created by a paying iCloud user and not an anonymous rando off the internet.
I'd be interested to know what downsides, if any, you see for a website to accept a private e-mail address like this. Do you have a legitimate complaint about these sorts of e-mails? Again, given that private relay isn't an 'anonymous e-mail service' (it's still tied to your iCloud account so spam, etc. shouldn't be any more of an issue) but merely an 'anonymous to the person you're giving the e-mail to' service.
If your actual complaint is 'if you insist on giving me an e-mail that you can revoke unilaterally making me unable to contact you against your wishes, and which cannot be associated with other user data from other sources to build a profile of you, then you're not worth having as a customer' then that's a separate complaint - and one that means I want nothing to do with your website.