The reason I'm saying use a VPN is because you're presumably visiting the site anyway, so leaking your full IP to them anyway. You can route your DNS lookups through what server you like, obviously. (Again, the privacy issue would be not that you're leaking part of your IP to archive.is, but to everyone in the chain of recursive DNS resolvers). You could use TOR too, I think even in this thread someone posted a TOR url for it.
Cloudflare do make the DNS queries from 1 of their 180 locations, so there is some information being passed through about where the request is coming from in terms of load balancing.
I'm not arguing that Cloudflare are doing the wrong thing by omitting ECS data in general. Just that site owners have a right to do as they like WRT people using their website and if that includes blocking Cloudflare, so be it. What he is doing is not legal (or at least grey area) in many countries so anything that makes his life easier is understandable IMO.
Also, ECS leaking does not seem like a real concern for the vast majority of people surfing the net.
Lastly I don't think Google own 4.4.4.4, did you mean 8.8.4.4?
There's a comment on one of the HN threads that gives a little more insight - basically it helps him combat abuse by routing requests to the closest server outside of the requesting ips area: https://news.ycombinator.com/item?id=36971650
Not sure how that argument really holds up to scrutiny but it's something.