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I've not tried alternative DNS services. I've heard Cloudflare say they are not storing data they get.

Yeah, we generally have to take such claims on trust - but so far I am not aware of any reason to think Cloudflare are lying. There's always time for that to occur though!
It is generally a good idea to use the fastest DNS available, so maybe you can try it out by pointing your Pi to Cloudflare. I see good speed increases from doing that vs. the default DNS from my ISP.

I've got it using OpenDNS at the moment. May try some others later