I haven't looked into DNS-over-HTTP/3 (DoH), but there's a secure-ish DNS mechanism already: DNS over Distributed TLS (DoT) [1].
The main criticisms of DoT are that it's hard to analyze for cybersecurity purposes. Except DoT is not encrypted end-to-end but only encrypted hop-to-hop. This means you can put a proxy checking for cybersecurity threats in front of your DoT traffic into or out of your network, if you need to. DoT can also be peer to peer, which can leverage a lot of benefits. DoH is client-server.
DoH has all the problems that come with HTTP/s and seems a good way to break one of the most relatively reliable systems of the Internet, in the name of convenience.