HTTP/2 was designed for high latency networks. It supports stream multiplexing and headers compression to significantly shrink page load time over the internet. It has very limited use on the local network. Regarding server-side push, it brings about as many issues as benefits and is often counter-productive. Hopefully it will totally disappear when Early Hints are standardized (addressing the same benefits without the issues).

That said, we still want to address server-side H2 in 1.9 to address the CDN type of workloads where the origin servers might be far away. But that's less important.

gRPC is also a good candidate for end-to-end HTTP/2 :)

Haproxy would usually be on the same network as your www servers so even without http2 the response time for the backend requests will be tiny compared to the client <-> haproxy roundtrip time.

The promise of http2 is to lower latency on the client side and having H2 on the frontend will help with that.

Fewer concurrent connections for the client if you have a resource heavy page. Http 1.1 still gives you stair stepped responses if you have a large chain of dependencies. Chrome halts concurrent connections around 10 or so.