It looks like you can install custom android roms (eg cyanogenmod) on Amazon tablets (eg https://download.cyanogenmod.org/?device=otterx - see other Amazon devices in the sidebar too)

True. But you can pick up pretty much identical tablets (my guess is down to the PCB level, as it does not look like Amazon has done much custom design here) at AliExpress cheaper than that, and if you look around a bit or ask, finding ones that either come pre-rooted or where the vendor is happy to do it for you or point you in the right direction is easy.

This is where these cheap tablets really shine - they come from distributors that generally really don't care. On the other hand this is also the downside: They also really don't care about updates etc..

The biggest downside, I think, is the state of open drivers for the GPU's on these ARM devices.

Not sure what you mean by locked down. The amazon app store has a fair number of apps in it and it has always been possible to sideload apps from external sources, even sideloading google app services and play store (at least in the fire phone and previous gen fire tablet, not certain about the new one). I don't think it's any more locked down than Google Android or iOS.

The Allwinner based ones should be easy enough to reflash with an alternative operating system.