DOH can be implemented directly inside of the web browser application, since those browsers are obviously already doing everything over HTTPS. So the browser developers just have to build in a DNS client. From their perspective, I would see that being much simpler. And deployment is as easy as the next browser application update.
Are you sure? My experience so far has shown that only dnscrypt is widely supported.
Nevertheless, surely they must had some kind of issue with the existing solutions when they started working on it.
As for DNS over QUIC, I was under the impression that said solution did not make use of HTTP.
I believe it's possible to do both direct DNS over QUIC and ((DNS over HTTPS) over QUIC).
https://android-developers.googleblog.com/2018/04/dns-over-t...
And for HTTP/1.1 it should use WebSocket for a persistent connection, but probably it already targeted HTTP/2.