As was pointed out elsewhere, fiddling bits with constant time guarantees isn't part of the C specification. You need a dedicated implementation that offers those guarantees, which isn't clang (or C, to be pedantic).
Fortunately you don't have to go 100% one way or the other. Write your code in C, compile and check it's correct and constant time, then commit that assembly output to the repo. You can also clean it up yourself or add extra changes on top.
You don't need to rely on C to guarantee some behaviour forever.
I agree that an extension (e.g. a pragma or some high-level operations similar to ckd_{add,sub,mul}) that allows writing code with fixed timing would be very useful.
But we have generally the problem that there are far more people complaining that actually contributing usefully to the ecosystem. For example, I have not seen anybody propose or work on such an extension for GCC.
The problem doesn't stop there, if you want to ensure constant time behaviour you must also be able to precisely control memory loads/stores, otherwise cache timings can subvert even linear assembly code. If you have to verify the assembly, might as well write it in assembly.
The cryptographic constant time requirement only concerns operations that are influenced by secret data. You can't learn the contents of say a secret key by how long it took to load from memory. But say we use some secret data to determine what part of the key to load, then the timing might reveal some of that data.
