Not the way a naive reading of your post suggests, no. If you add a random amount of time, the timing attack still works; it averages out, the same way network

Not the way a naive reading of your post suggests, no. If you add a random amount of time, the timing attack still works; it averages out, the same way network jitter does. What you need to do is make every request take the same amount of time; randomness is not helpful here.