"Instead I'd produce a restricted C compiler that only accepted constant time programs, and mapped directly to a limited set of machine instructions."
This has more potential just because it could be licensed in embedded sector to reduce work on things such as WCET analysis for hard-real-time. That money and uptake can feed back into the compiler's development for things such as constant-time-preserving optimizations. I haven't seen one of these, though. They typically do timing analysis manually (i.e. specific tool) since they have to assess the algorithms themselves, interactions with other components, and so on. Neatest thing I found looking into that was a formally-verified WCET analyses that was built on top of CompCert compiler.
Oh wait, just found something: