So that's why 1.1.1.1 DNS failed.
I noticed around the end of the day (eastern USA) that I wasn't able to get to a bunch of Internet sites. I just assumed that systemd-resolved had b0rked itself and I'd fix it later, but when I came back to the computer in the evening, everything was fine. I use the 1.1.1.1 DNS resolver, which is operated of course by CloudFlare. I guess I'll just have to set up my own DNS resolver that talks directly to the root servers.