"What does it mean though for a GUI to be keyboard-driven?"
For literally decades after they were state of the art, people who used old-school mainframe TUIs would still swear by them.
I think the key difference is that they would buffer inputs. You could be on the "Receipt" screen, and know that if you needed to enter the code for some rebate receipt or something, you could hit F6 - F2 - down - down - down - {start typing code}, and even if the UI took multiple seconds to actually get to the point that the code was being entered, the input would be kept.
Normal GUIs have discarded this for fairly good reasons. Especially when it comes to queuing mouse clicks. If you get down to the nitty-gritty of what it means to queue a "mouse click" it get very conceptually complicated. But you could create a GUI toolkit that still had the same idea integrated into it for a keyboard interface, if you have well-defined points where you are and are not accepting keyboard input.
I would submit that "keyboard-driven UI that is as intuitive as clicking with a mouse" is actually the wrong direction to cut on. You need to beat the conventional mouse GUI completely in some other dimension, one where it can't even compete. This is the only one I know of that has historical precedent.
The downside is that on a desktop I'm pretty sure you're looking at a completely new widget toolkit. Trying to force this on the current ones is going to be hard and could even be impossible.
If I were going to create a new web frontend framework, this is what I'd focus on to have an actual new feature that isn't just a respelling of the current set of features. And I can't guarantee that it won't also be missing something crucial, though, I think everything you need is actually there. You could even have modern nicities like, store the entire GUI field state transition diagram statically at the beginning and as the user types ahead of the GUI's current state, show them a rendered view of where they are going, e.g., the key sequence I showed above could be "[Rebates] -> [Rebate Receipt] -> [Name Field] -> [Address 1 Field] -> [Address 2 Field] -> [Rebate Code Field] -> [Entering {'A8F2-132A-9283'}]", live updating as it goes so if the user can just bang that out even while the first screen hasn't loaded they can even be sensibly editing the text input they're going to put in.
Combined with something as simple as making sure everything always has a keyboard shortcut in a defined format, which is conspicuously displayed every time you change something, e.g., every time the user clicks the Rebate Code Field you can also display something like "Ctrl-F R" or something so they also learn how to drive the screen with a keyboard and you would, if nothing else, have something different than what's out there today.
Anyone who wants to, steal this idea. I'm not going to do anything with it. I don't see this as something you want in every app necessarily, but people who spend hours a day in the same app for whatever reason would be confused for a week, then I bet you couldn't pry it from them with a crowbar.