Similar results from me. 1.1.1.1 for DNS works for a while, then all of the sudden, sites can't be found. Change to a different DNS provider and it suddenly starts working.
1.1.1.1/1.0.0.1 is just a public DNS service hosted by Cloudflare. This is act as a encryted proxy solution that has two parts, one on Apple servers that can only see where you are coming from, the second part at Cloudflare that can only see where you are connecting go to. This makes it completely private as all the traffic is being encrypted so third-parties have no clue what you are doing. Apple has no clue about you destination as all they can see as the destination is the next hop in the chain which is the relay server hosted by Cloudflare. And Cloudflare has no clue where you are coming from as this information is being obfuscated by Apple before it's being relayed to Cloudflare.
End result is completely private all the way through the chain for every party that is involved. That's not to say that Apple won't be capable of putting on-device scanning like they do with CSAM. As the scanning is being done on-device it's done before the traffic is being encrypted and heading towards the relay servers. At that point it wouldn't be private as a result of Apple having insight on everything that is going on completely defeating the purpose of how Apple have designed iCloud Private Relay. At that point Apple could simply have a encrypted proxy directly to Apple and not have the two chain loop involving Cloudflare to make it completely private.
When it comes to 1.1.1.1/1.0.0.1, Cloudflare is a great service provider and their public DNS servers are considered to be some of the best in the business. You shouldn't use their iOS app, that one implements a VPN service to catch your DNS traffic which is horrible but the only way Cloudflare can make it happen in a easy way for users on iOS and iPadOS.
You should just grab a iOS/iPadOS/macOS mobile config file:
GitHub - paulmillr/encrypted-dns: DNS over HTTPS config profiles for iOS & macOS
DNS over HTTPS config profiles for iOS & macOS. Contribute to paulmillr/encrypted-dns development by creating an account on GitHub.
Grab the Cloudflare HTTPS mobileconfig and install it on iOS, iPadOS or macOS and you will have encrypted DNS towards Cloudflare directly. No VPN nonsense or anything going on like it is when using the 1.1.1.1 app from the App Store. Works so much better. If you don't like it, simply remove the profile. It's as simple as that.
It's better to use the HTTPS profile over the TLS profile. Some public and corporate networks might be blocking TCP-853 which is required for DNS over TLS to work. 99,99% of all networks will allow for outbound TCP-443 traffic as that's needed for Internet/webtraffic to work so DNS over HTTPS will work in almost any scenario so it's a safer bet.