Robert - a speaker could be designed to be driven by 2 amps as you suggest, or the effectively identical single stereo amp where each channel drives one XO. Or is it identical? I experimented with that 'vertical bi-amp' vs 2 separate amps vs one channel driving the whole speaker in the days of the CS3 (1983) which had dual inputs. The results from those very similar situations were different enough that we (Jim) would have made different XO circuitry decisions to attain the desired linearity. The reasons are real, but below the threshold that common 'wisdom' deems sonically important.
There lies the far more gross problem of user misapplication of double inputs: different amps, cables, etc., but the problems exist even with the perfect implementation that you suggest.
The global transducer that we call a loudspeaker is a complex, reactive device. When drivers accelerate, move air, resonate and so forth, reactive electromotive forces with their electrical counterparts seek equilibrium withing the entire system. The positive signal and negative coupling reference are bonded at some point. In Thiel speakers that point is at the single binding post pair. If that point moves (such as moving it to separate amplifiers) many aspects of the global transducer are altered. Even if amps, cable types and lengths remain identical, enough has changed to require design modifications (in Thiel's design framework).
An advanced DIY builder might devise the required modifications to make the new system right. But that is a very long shot, and would only apply to the particulars of those particular driving components. The short answer is that the complex interaction of the variables makes the new configuration into a new design project.
Your query from a different slant raises some very interesting avenues of improvement. My journey has demonstrated that separating driver and even individual circuits from each other (in space) serves to simplify and clarify the signal propagation through the circuitry. Various distortions, primarily from field coupling, are significantly reduced without significant change to the primary (driver-XO) dynamics. The biggest deal is getting the inductors away from the driver motors. Note that any of our products that have separated XO panels that are mounted some distance from the drivers are better sounding products. The 2.4 with its 2 XOs mounted in the cabinet bottom meets those criteria. Compare the 2.4 to the CS2.2 XO mounted behind the woofer. Simply moving the 2.2 XO outboard fixed a major problem of what sounds like woofer bottoming. The splat at high power vanishes via cleaning up the field interactions. I am pursuing such improved layouts without the insurmountable baggage of multiple inputs or amps.