If you use the crossover in the head unit, then you aren't sending that part of the signal to the amp. (Whether or not that allows a more narrow signal to be "cleaner") someone else might be able to answer.
Have you checked to see if you can use both? See what the frequency range is on the speakers you are using. You might be able to use the high on one and the low on the other.
If you're someone that disconnects your battery often for maintenance, the amp's crossover is mechanical and wouldn't need to be reset every time.
It won't save you money. Yeah you lose the fader ability from the head unit, kind of depends on your setup.
I did something like that for a different reason. I have 3 amps. I have three pairs of outputs on the head unit. One amp for subs, one amp for the front components (2 pairs of nice and loud), and the third tiny amp for the rear mids that are there because I have the space, had the equipment, and think they help at mid volume. I ran RCAs from front and sub channel to under the drivers seat, and added in a potentiometer (bass knob) for each. After the speaker knob, I split it and ran two sets to go one for each speaker amp. I lose a little fader, but I'll never really notice it, and this gives me the option to run subs independently at full volume to be obnoxious, and can turn the mids and tweeters all the way up or down independently. I leave the head unit on near high (for most of the time), and can adjust it all a bit easier and incognito from the seat without going through the head unit.
