So, I have looked around and refreshed what I've read, and what I've saw. Let me try to summarize.
In short, the controller you are referring to as defines what you can send to the screen and what not, in other words, it gives you a capability ceiling, but the capability is not inside the controller. In your words, it defines the capability maximum for "waveform table processing".
On the other hand, the controllers you have are sitting ducks without these waveforms (which my mind conventionally renamed them to wave functions, well because they're waves, and they're defined as functions, and my brain is all hazy because everything I read, sorry about that).
These waveforms allows us to reach from a state to a state, generally in page scope, so you generally refresh the page completely. I understand that partial refreshes are also possible, but I don't know how they're implemented, exactly. My Kobo does it rather elegantly, but it might be pushing whole-screen updates too. Anyways.
[0] contains a PDF file which details how to process a waveform table, probably sold as a part of a screen by Waveshare. It details how to interpret the file, and how to make use of the data in the flash file, and drive the screen in most efficient/correct manner. It details gray levels, some special functions for maximizing blackness, etc.
[1] contains the article which brought me to the entrance of the rabbit hole. They give the basic details how they're software defined and controller can use many available waveforms, and you need to license them separately from the controller itself.
[2] is a product page for an eInk display, and it briefly mentions that the image quality you can get it both dependent on the controller and the waveform file you gonna get.
At the end of the day, yes you gonna need a panel controller to talk to your display properly, but what you can do and can't do with it is codified inside the waveform file you get with your device, and it's decoupled from the controller itself.
I guess it's possible that the controller you have can be too primitive to interpret or execute the waveform you have at hand, but vice versa is possible. i.e. your waveform is more primitive than the controller's capability ceiling, and despite having the best controller, you can't get most of the both your controller and your screen.
As a last note, [3] contains a reverse engineering effort and project for these waveforms, which I was not aware of before.
[0]: https://www.waveshare.net/w/upload/c/c4/E-paper-mode-declara...
[1]: https://goodereader.com/blog/e-paper/e-ink-waveforms-are-a-c...
[2]: https://www.eink.com/product.html?type=productdetail&id=1
[3]: https://hackaday.io/project/21168-fpga-eink-controller/log/5...