I can still determine timing information by measuring how long it takes to execute a program. The only way to prevent this is to enforce constant-time programs

I can still determine timing information by measuring how long it takes to execute a program. The only way to prevent this is to enforce constant-time programs by delaying a response until a specific amount of time (see constant time comparison functions in cryptography). That's not feasible for many applications, especially operations on a latency-sensitive critical path.