Well designed crossovers can be very good, but the problem is many are not well designed. There's also the problem of variation in the mechanical and electrical properties of the drivers.
Couple that with the typical crossover point between midrange and tweeter being right about where the human ear is most sensitive to weirdness, and the result is what you find on the market- lots of mediocre speakers and some down right bad ones.
There are no true full-range drivers, but electrostatic speakers can come about as close as you can get. They are typically much less sensitive than speakers with dynamic drivers and crossovers and require large surface area to produce even mid bass (forget about low bass). They require equalization to flatten their response and they are "beamy", limiting the sweet-spot to one person and requiring careful room placement, typically at least a few feet away from the wall behind them.
The large size, poor low frequency performance, and low sensitivity make them relatively unpopular. They require that you arrange the room around them, so the WAF is relatively low. If you want some bass to go with them, you have to add bass cabinets and maybe an active crossover and a second amplifier (unless you go the hybrid route like Martin-Logan).
All that said, there isn't anything out there that compares to the sound quality of ESLs (which is exactly why you see so many ads for speakers comparing their sound to ESLs!).
"If a crossover is designed correctly, then shouldn't the resulting response be flat? If so, then why the whole deal about full-range drivers? "
Full range speakers is generally a niche, just like horns, esls, etc. As with everything, each type has to compromise.
As MreHost stated, the main claim that FR enthusiast assert is the lack of a crossover. Well, this is a debatable topic that could go on forever between parties. As well as lack of a crossover in the sensitive range, also phase response is 'generally' bettter on a FR. However, this is not true when considering multi-way designs using true transiet perfect 1st order acoustical slope crossovers, which have very simalar phase response. And, most FR drivers have worse phase response than properly designed TP multiways. Of course, even the audibility of phase distortions of high order xovers(2nd order and greater) is debatable. LOL.
Dynamic FRs offer some very substantial negative attributes though: high intermodular distortion and poor power response, through the intended bandwidth. One thing I do not personally agree with MreHost about is ESL vs. other designs. Thier are applications using converntional cone drivers/planars/hybrids etc that have very similar transient decay characteristics to an ESL(the main benefit of an esl), but with BETTER power response(off axis response). I do admit, however, that these are rare for numerous reasons. So, in GENERAL....ESLs offer better 'clarity' than other designs. MreHost probably meant MOST, but so many are inferior, ONLY considering spectral decay, that this would be an accurate general statement. As far as crossovers being located in the sensitive ranges, this is avoidable with very careful and selective design. But the added costs of achieving this are usually prohibitive, ESPECIALLY in retail products, and substantial skill is required to acheive this in the DIY route. Also, keep in mind that EVERY paramter'w importance is dependant on the intended end user's preferences. Many people specifically prefer speakers with worse spectral decay response, as this adds sometimes 'life' and 'texture' to the sound, which in reality is coloration. In this case, the last speaker such an individual would want is an ESL or transducer system with similar spectral decay behaviour. Also, such speakers are ruthless, requiring exceptional recordings..if you wish to achieve a virtual performance in your listening room. In this aspect, I don't know a single dynamic FR driver that exhibits very good spectral decay response. This is reserved for ESLs and Mangers, in the FR neighborhood. But as I stated, this is a preference. You should sample all types, and decide your preferences. That is what is really important anyways. Targeting ideal measurements alone will lead you to a DEAD END.
The more you try to do with multiple drivers in a box the more compromise and usually the more asked out of a crossover to make it sound right. I have some close to full range drivers that are pretty old. At least 40 I would guess. A 15" full range with a tweeter in the center on a bracket. There is a cap in there so it is relly a 2 way with a 6db high pass. A pair of these sound great and have the alninco magnets so I can get plenty of volume with a SE 6BQ5 amp. I have built horns for single driver no crossover use and some were very nice. Problem is that it is a middle of the road sound. It can't go too high and without a folded horn the bass would be very poor. They do fantastic in the area where we here 95% of what we listen to and some are better than others and have greater ranges. I still get back to the 2 way. If you have a woofer that can go from 30 to at least 8K that falls off quickly with no spikes a 6 db high pass works fine. Takes some fine tuning but will still work. Complecated crossovers are also power hungry so the simpler the cross the more efficiancy which is inportant for SE tube amps. If you have 100s of watts to play with just getting the crosses in the correct places and not using cheap componants ought to work. Like in everthing else the way you get someplace will decide what you need to get there. I need to keep things simple because of low power. There is no answer that is correct for all situations for dealing with passive crossover design.