Why is addition as fast as bit-wise operations in modern processors?
Addition is fast because CPU designers have put in the circuitry needed to make it fast.  It does take significantly more gates than bitwise operations, but it is frequent enough that CPU designers have judged it to be worth it.  See https://en.wikipedia.org/wiki/Adder_(electronics).

Both can be made fast enough to execute within a single CPU cycle.  They're not equally fast -- addition requires more gates and more latency than a bitwise operation -- but it's fast enough that a processor can do it in one clock cycle.  There is a per-instruction latency overhead for the instruction decoding and control logic, and the latency for that is significantly larger than the latency to do a bitwise operation, so the difference between the two gets swamped by that overhead.  AProgrammer's answer and Paul92's answer explain those effects well.