They're cheap but not free, especially at the front end of the CPU where it's just a lot more instructions to churn through. What the branch predictor gets you is it turns branches, which would normally cause a pipeline bubble, to be executed like straightline code if they're predicted right. It's a bit like a tracing jit. But you will still have a bunch of extra instructions to, like, compute the branch predicate.

Worse, IMO, is the never taken branch taking up space in branch prediction buffers. Which will cause worse predictions elsewhere (when this branch ip collides with a legitimate ip). Unless I missed a subtlety and never taken branches don’t get assigned any resources until they are taken (which would be pretty smart actually).