Unit tests that throw random numbers into the code and check execution time is always a constant number of clock cycles should catch nearly all timing side channels.
What you really want is https://github.com/agl/ctgrind or something similar.
And it’s worse: that your code is constant-time in the absence of an adversary doesn’t guarantee that it is so in the presence of one. You will have to stress test your code with code running in other CPU threads that flush various caches, switch CPU modes, do DMA, etc.
Having said that, running with random inputs isn’t a bad idea, if only because you have to get confidence that the CPU manufacturer’s timing data is correct.