One good thing about an outboard harmonizer is that it still works after the style stops. It remembers the last chord.
One REALLY BAD thing about PSR harmonizers is that there is no light indicating whether it is engaged.
One great thing about Korg PA4X harmonizer is that you can assign Harmony Volume to a footswitch and, with a momentary pedal, harmonizer only engages while you hold the pedal down.
It actually is always engaged, but the pedal reduces the harmony volume to zero, instantly.