Would it not be better to crossover before amplifing the signal, or am I missing something?
better is up for debate but for an active one you need 3 amplifiers per channel for a 3 way system vs only one for passive. Good amps aren't cheap.
Speakers evolved as full range devices so you would have to change the whole culture of high end. Few speakers are designed so you can easily bypass the internal crossover.
Plus why can't they separate the highs, mids, and lows during the recordingthey could but what medium would you use and where is the market? Up until the dawn of 5.1 surround and hi-rez recordings nothing could handle 6 channels. The average consumer won't accept SACD and wants mp3. Do you think they will buy such an elaborate system?
what frequencies would you choose as cutoffs? And why 3? Maybe I want a 2 way or 4 way system, and every speaker designer would want a different cutoff for their drivers of choice.
I think you are going to have to make your own recordings and build your own speakers.