AES is actually a good example of why this doesn’t work in cryptography. Implementing AES without a timing side channel in C is pretty much impossible. Each architecture requires specific and subtle constructions to ensure it executes in constant time. Newer algorithms are designed to not have this problem (DJB was actually the one who popularized this approach).
Reconcile this claim with, for instance, aes_ct64 in Thomas Pornin's BearSSL?
I'm familiar with Bernstein's argument about AES, but AES is also the most successful cryptography standard ever created.