I don't think it needs to be so dramatic. You could have a proof that the algorithms don't contain data-dependent logic, and perhaps ensure that no data-dependent instructions are generated; and that all branches have the same number of instrution-cycles (adding padding if not). You'd then simply rely on the architecture-specific instruction timing differences to be respected by the compiler.
Would something like this guarantee that no side-channels are possible on any architecture? Perhaps not, but it would still get you most of the way there.
I'm not so sure. It's pretty hard to know how many cycles a register load instruction will take if there is a cache, or worse a cache hierarchy. That's why I think it will be a lot easier to have a proof on simple cache less designs...
Your version is likely good enough in practice though.
What about power draw? Maybe that doesn't count as a side channel (I'm not a security guy). Afaik, CMOS transistors mostly draw power on switching state, so a normal adder adding 0+0 or 1+1 is visible in the power draw.
I googled a bit and found Sense Amplifier-Based Logic (SABL). Super interesting :^)