Don't know about AESA, but PESA's are usually built on klystrons or magnetrons, which are strongly limited in frequency bands.
You're wrong in that.
Yes, AESA can split beams, just like the Fly-By-Wire system on F-22 theoretically allows to incline thrust vectors and ailerons in opposite directions, but why do so in conventional flight, or (for radar) if it allows fast scanning just like PESA? The actual splitting is needed only when handling really large number of targets. Therefore, NO range decrease of AESA in comparison to PESA, except ones to enchance LPI capabilities or due to less desing experience (like for russian ones).
The only advantages of PESA are less heat, less expensive production and slightly larger FOV.