PA speakers sometimes carry "midbass woofers", which might sound like an OK specification for bass, but actually isn't. As you noted, PA speakers are usually/often used as "satellites" or in a line array, and are therefore dispensed with reproducing the low end. Midbass woofers are intended to be used from 80-200Hz up to approximately 2.5-3.5kHz, where the tweeter then takes over, with subs handling frequencies below. They would incur overexcursion if used "as is" with a bass guitar signal, and as agedhorse noted, their duty is to reproduce well (efficiently, powerfully and with as little colouration as possible, although pro audio is not as "hi-fi" as true hi-fi) a narrower frequency range. (Some speakers are intended to be usable alone, and they might carry woofers that are a bit more similar to bass guitar woofers, but might sound "dull" in the high mids).
In bass speakers, woofers are not low-pass filtered usually, and the tweeter is merely "added" over the woofer, with high-pass filtering only on the tweeter; it's not hi-fi by any means, but bass amplification isn't hi-fi usually. Reproduction in the high-mids will indeed be somewhat "unpredictable" due to the interaction between the woofer and tweeter (which also depends on the listening position), but this arrangement allows the user to dial in the amount of treble by dimming the tweeter as required. In PA speakers, the high-mid voicing of the woofer is not very important (although it shouldn't be too "non-linear", to make it easier to cross it over with a tweeter).
My opinion on the matter is that bass guitar amplification (and why not, guitar amplification) in many cases would greatly benefit from a more "PA" approach, with flatter response and proper crossovers and tweeters, and with mid/treble voicing taken care of by modelling and/or filtering and equalisation rather than by the woofers themselves. This would come with one big benefit: better polar dispersion, with the whole band, stage and possibly audience hearing the bass a bit more as it should be heard (and also programmability, with the option to tweak the response); on the other hand, this means that bassists need to approach their tone shaping and monitoring differently, since the narrow dispersion of mid-high frequencies of traditional cabinets means that bass players can dial in more high mids than appropriate to benefit from the added definition. I believe that this only works on stages with proper PA support; if an amp is expected to provide sound to the audience, this works poorly (and in fact, when the band's amps provide the sound to the audience, the sound is usually subpar and very position-dependent).
But after all, we are all seeing this trend, with bands investing in good PA speakers and then musicians just plugging in their Kemper or whatever modeller or just a regular preamp/DI into the mixer, and relying on IEMs or other personal monitoring. It is both more convenient and can sound better.
