The idea that middle boxes need to be aware of protocols is incredibly destructive to the end-to-end principal - the critical property of IP networks that enables a rapid pace of permissionless innovation. Middleboxes are evil incarnate.
Fortunately, academic research as well as operational experience has shown that TCP and UDP provide the primitives required to build any protocol. QUIC, which you mentioned, is an example of this. Google has made the explicit - and probably wise - decision to keep protocol state encrypted. While there would be some nice optimizations possible if routers could be aware of things like 'this session is closed now', the risk of middleboxes making it so that the protocol can't continue to evolve is too great. The solution is to specifically prevent middleboxes from having the knowledge needed meddle in an 'intelligent' way.
The last statistics I saw from Google were that 95% of networks don't cause problems for QUIC (i.e. QUIC connections are as fast or faster than HTTP/2). The main issue I'm aware of is not so actually de-prioritization but rather rate limiting (it's often implemented as a quick - and very very dirty - DDoS mitigation technique). Google's stated solution is to fallback to HTTP/2 based on measurements of user experience on a per-AS basis.
Networks which try to be "intelligent" rather than just provide adequate bandwidth will tend towards providing a crappy user experience. Solutions which optimize for well behaved adequately provisioned networks and have an acceptable-though-degraded fallback seems like a reasonable way to promote progress.
SCTP has an RFC-defined UDP encapsulation which will deal with many evil networks.