the amplifiers will be much smaller in power rating and will be economical.
Not really. If you have say a 100 watt amp for a three way speaker, you are still going to need an amp very close to that level for the woofer, and you will need 2 more amps to boot. Yes, the midrange and especially the tweeter will not need as much power but the quality still has to be there. Even if you could get by with 3 amps that had 1/3 the power it would still cost more than a single higher powered amp. Going up the power chain of a power amp line doesn't double the cost when you double the power . It is a fraction of that.
I agree that a well executed system with an active crossover has many advantages for the reasons listed, but saving money is not one of them. Quality amps of any power level are not "economical."