The DNS on my network is my router, but that has 1.1.1.1 as its upstream resolver, and private relay is working for me.

I wonder if this is an interaction with the way the cloudflare app sets up 1.1.1.1. - I guess as a VPN? — rather than a service incompatibility

Yes, it must be the vpn set up. Though no issues are present when warp is active.