archive.today isn’t associated with archive.org. It’s run by one guy who has a grudge against Cloudflare and gets angry when you use their DNS or anything that proxies your requests through Cloudflare, include iCloud Private Relay. You end up on a page that looks vaguely like a Cloudflare challenge, even though it’s his own doing and has nothing to do with Cloudflare.
If you’re on iOS, just open it in any browser other than Safari. Other browsers don’t get provided through iCloud Private Relay.
to be fair, I'd block visitors to my site if serving those visitors could result in legal action that would force me to take the site down. I hope I'd be a bit less juvenile about it, but it's not my site.
You would need iCloud Private Relay enabled to encounter the issue. Even then, you don’t get a say in which edge provider you get routed through: it may be Cloudflare, or it may be someone else (e.g., Akamai). It’s up to Apple.
The trouble is that most people who have it enabled don't realize. If you're paying for iCloud+, it's likely enabled.
I'm not suggesting anyone should be complaining to you, though--I'm just explaining why some people encounter issues visiting archive.today links and providing them with a workaround (disable iCloud Private Relay).
Or maybe to your ISP?
Or VPN host if that’s involved?