DoH3 means running DNS over HTTP over QUIC over UDP. Here QUIC does both session management and encryption.

In both cases, we are running a simple application protocol (DNS) over other protocols that handle the Internet-level problems I raised, so all is good.

The problem is with running your application protocol directly and strictly over UDP and nothing else.

And related to sessions, there are two things. For one, in reality today, you typically do a whole host of DNS requests even to load a single site (many common sites have upwards of 20 domains they use, and that's before loading any ads). So having a persistent session to send all of those requests on would not change much, even if it's not technically necessary. Secondly, even if you really want to avoid sessions, you then still need some other mechanism to prevent source IP spoofing.

Any protocol which allows a host to send a small request to a server and cause that server to send a large response to the src IP of that request is a major problem for the health of the internet. Requiring a handshake to solve this is one simple way to avoid the problem entirely. DNS implementations have had to find all sorts of other mitigations to address this (I believe they now typically don't allow responses more than a factor of 1.something larger than the request, or something like that? Which of course brings in all sorts of extra problems and unnecessary traffic)