A line level signal is amplified full-range, then sent to a crossover where part of that amplified signal is deliberately filtered away for each driver.Why do you assume that a crossover ‘strangles’ the signal? It’s more accurate to think of it as a device that sends the relevant part of the signal to the right place.
Which rather begs the question: Why bother amplifying a full-range signal? Why not filter it at line level first, so that amplification won't be wasted? It seems like a pretty obvious course of action to me...