Again the solution seems to be in that the Apple III was built around Apple II Hardware. According to the Apple III FAQ:
So it seems pretty clear that third party cards could make the Apple III just as capable as similar cards required to make the II Plus as powerful.
Now that I am clear on exactly what the 65c816 was doing vis-a-vis hardware, I'd have to say that the III had as much right to be called a II as the IIGS. The only difference being that the III did everything it could to cripple support for the Apple II mode out of the factory and the IIGS did everything it could to facilitate it. Nevertheless, with the same kinds of add-on cards required to provide the same functionality on an actual II Plus, the III was a perfectly capable II Plus (indeed even a IIe), with the same processor running at its core. Had the Apple III stuck around longer, I wonder if more developers would have attempted to circumvent Apple's debilitating efforts. In researching this it seems the main reason for the III's failure was a complete lack of advance developer support (a problem Jobs astutely avoided with the Macintosh). Even if the hardware had been trouble free, the lack of available software and lower price would have likely kept the Apple II alive for many more years anyway. Who knows, eventually Apple might have merged them into the IIGS anyway, as Apple realized the folly of trying to separate the two systems.