Archive.* sabotages their DNS records when Cloudflare queries for them. They don't like that Cloudflare doesn't do EDNS forwarding so they broke their service for people using 1.1.1.1.
That said, I have the same problem. Even hard coding the IP address I resolved through Google doesn't seem to work. I'm guessing their sabotage may have backfired and is causing issues beyond their intentional scope?
This just helped me realize why I couldn't get to archive.today anymore -- however, for me, both Google DNS (8.8.8.8) and CloudFlare DNS (1.1.1.1) resulted in either infinite captcha loop or timeout.
I had to switch back to my ISP DNS to have connection successful.
I did not realize that choice of DNS resolver could effect access to a website like this. I thought DNS was boring stable technology. The error conditions weren't even DNS failure (which I would also find surprising from Google or Cloudflare), but that server timeout, or weirder infinite captcha loop.
DNS was ISP, not 1.1.1.1, and I get the same behaviour after switching to 8.8.8.8.