*If* a non-Disney bus should pull up to your stop, or a bus that is not equipped with a ramp or a lift, you can either ask the driver, or one of the Transportation CMs to call a bus that can accommodate your ECV.
I am not sure if there are still as many "non-Disney" buses running now; when we were there at Thanksgiving through the first 2 weeks of December, at any given time as many as 50% of the buses were "non-Disney" buses; they were all motor coaches (so the same style of bus as the Magical Express used) and were NOT being driven by CMs. I have never seen so many different bus companies in one spot!
Overall, the Transportation CMs were right on the ball about making sure that we never got a bus that wasn't a "Disney" Bus; there were only a couple of times in those 2 weeks when we had to wait for a "real" Resort bus. One time, we even saw a Transportation CM wave off the bus that was headed for us at the last moment, and then wave in a Disney bus to our stop!
Yes, they should. Most of this thread is from pre-covid, but the lack of posts indicating a change leads me to believe the shuttles are similar now. It sounds like most are coach-style buses with a lift, as opposed to the commuter-style WDW buses with a ramp.