DNS over HTTPS (DoH) is entering the last call (right now Working Group, soon IETF wide) stage of IETF standardization. A common discussion I have about it basically boils down to "why not DNS over TLS (DoT)?" (i.e. work that has already been done by the DPRIVE WG). That does seem simpler, after all.
DoH builds on the great foundation of DoT. The most important part of each protocol is that they provide encrypted and authenticated communication between clients and arbitrary DNS resolvers. DNS transport does get regularly attacked and using either one of these protocols allows clients to protect against such shenanigans. What DoH and DoT have in common is far more important than their differences and for some use cases they will work equally well.
However, I think by integrating itself into the scale of the established HTTP ecosystem DoH will likely gain more traction and solver a broader problem set than a DNS specific solution will.
Using HTTP also has advantages at the protocol level. Both DoH and DoT use TCP and therefore need to implement a multiplexing layer to deal with head of line blocking. DoT uses the Message-ID as a key (inherited from DNS over TCP). DoH addresses this via HTTP/2's multiplexed stream approach. The latter is a richer set of functionality that deals with a wider set of edge cases. For instance HTTP/2 includes prioritization between streams, the ability to interleave different responses on the wire (e.g. to pre-empt an AXFR transfer), and flow control to limit the number of requests in flight without resorting to head of line blocking
As HTTP/2 evolves into QUIC, DoH will benefit from being on the QUIC train without any further standardization. Likely all it will take to deploy DoH over QUIC will be a library upgrade. This will be a very important performance improvement for the protocol. Both DoH and DoT are essentially tunneling datagrams over TCP. Anybody who has run a VPN over TCP knows that's not optimal because packet loss in TCP delays the delivery of many other packets within the connection. This creates an unnecessary slowdown when unrelated things, such as different DNS responses, are carried in the same TCP flow. QUIC addresses that issue and DoH will definitely benefit from the free upgrade.
The big picture potential of DoH emerges only when you zoom out from the DNS context. When you integrate DNS into HTTP you don't just use HTTP as a substrate for DNS transfers but you also integrate DNS into the gigantic pool of pre-existing HTTP traffic.
HTTP has been working for the last few years to reduce the number of connections necessary to transport HTTP exchanges. Every time this happens the transport gets cheaper due to fewer handshakes, faster due to better congestion management, more private due to reduced use of SNI, and more robust to traffic analysis because the pool of mixed up encrypted data becomes more diverse.
Roughly speaking, this is represented by the evolution of HTTP/2 and its extensions. Highlights include HTTP/2 itself (mixing slices of different HTTP exchanges together in real time on the wire), Connection Coalescing (allowing some classes of different origins to share the same connection), Alternative Services (allowing HTTP to control routing and therefore increase Connection Coalescing), the HTTP/2 Origin Frame (making management easier to fine tune Connection Coalescing), and the upcoming Secondary Certificates (allowing a connection to add identities post handshake and therefore, you guessed it, increase Connection Coalescing). This system forms a positive feedback loop - new data enjoys the benefits but it also increases the benefits to all of the existing data. Adding DNS to the mix is a win-win.
The last important bit to touch on is more speculative. One of the newer HTTP semantics is push. Push allows an HTTP connection to send along unsolicited HTTP exchanges on existing connections. By defining an HTTP transport for DNS it can logically be pushed by servers that have reason to believe you will soon be interested in a DNS record you haven't even requested yet. This is an exciting concept, but it also comes with some tricky issues around security and tracking that need to be fully explored and standardized before it can be deployed. Nonetheless, architecting a secure DNS around HTTP enables that possibility down the road. The HTTP and DNS communities are just starting to look at the issues of "Resolverless DNS".