As noted above, there's no particular reason not to do what you're talking about, except expense and complexity.
But conceptually, you're absolutely right: DSP for room correction includes digital EQ and filtering, which is essentially the same thing one would use to electronically manage the crossover between different speaker drivers. Even more so, bass management when using a subwoofer already
is what you're talking about, except what you're talking about is extended to the crossover between all the drivers.
There already are active crossovers available, both digital and as
@restorer-john says, analogue.
What we don't yet have is a series of more mainstream, mass-market (or at least audiophile/DIY mass market) single-box devices or software packages that can do room correction, bass management
and also total active crossover duty. That last bit is the tough one, simply because every speaker requires a different crossover network, and no software or one-box device could properly manage a speaker unless the manufacturer published detailed technical specs on the crossover (and the crossover inside the speaker could easily be bypassed).
Also keep in mind that lots of speakers have dual connectors on the back, but often this is for bi-
wiring rather than true bi-
amping. In other words, even if you disconnect the straps that connect the two sets of speaker binding posts to each other, this does not disable the internal passive crossover circuitry in most speakers.