I think doing this concept properly is as much of a software project as a hardware project.
Having used an e-ink device heavily for over a year, including both local (Termux) and remote (SSH) system use via it and a keyboard, I'm conveninced that for text-heavy work it's fully functional, and would probably be viable with a sufficiently-thoughtful GUI which adhered to the design principles and respected the imitations of e-ink:
1. Persistence is free.
2. Pixels are cheap
3. Paints are expensive
4. Refreshes are slow
5. Colour is limited to nonexistent.
6. Pagination over scroll.
7. Full refresh (of page or portion) over pan.
8. Reflective rather than emissive.
9. Minimise animation.
10. Line-art or halftones over shade gradients (images).
For e-ink, it's the electrophoretic display itself which imposes lag. It's a bit like pushing a wet noodle or a heavy arm. No matter how fast the graphics card is responding, the physical display simply can't keep up and iposes inertia on changes.
Again, for text-heavy applications and reasonably static imagery, this isn't much of a problem.
Current displays can cycle between refresh modes, and it's possible to trade image quality for response time. Presently that's only possible for the whole display at a time, but I might imagine it would be possible to zone a display such that a portion is at a high-fidelity mode and another at a lower-fidelity, higher-refresh mode.
If the idea is for a low-distraction device, I'm not sure that's actually the direction you'd want to go in, however.
If we go really old, we start getting into really slooooow phospors. It would take multiple seconds for a blinking cursor to fade.
Or punch cards.
Or toggled programs.
Or jaquard cards.
Or ...
The discussion above was of the 1990s. Displays were graphical and bitmapped, not text. I remember when dragging and resizing windows with contents visible became A Thing.
And yes, I used computers in the phosphor age. And just a tad during the teletype period. I don't think either applies, though e-ink does have some characteristics of phosphors in terms over overall response rate.
The main distinction though is that phosphors painted in a line, not a full-screen refresh (though it might be interesting to see just how e-ink does refresh in super-slow motion), and then decays.
For e-ink, the characteristic is that higher-quality refreshes take longer, and that a full flush (blacking then blanking the full screen) creates a marked flash. It's much better on recent high-end displays than on, say, a ten-year-old Kindle. But there's still some of the effect visible.
Much less a problem when paging, or when there's a single cursor-point-of-change when typing text. More a problem with complex dynamic displays.
All these speeds are in a dithered 1-bit mode though. As soon as you start playing around with grayscale things starts going slow (you essentially need to display N levels of grayscale as N frames). The system I'm using is capable of updating parts of the display in 1-bit mode and others in "high-fidelity" 16 levels mode. This means that I could e.g. tell it that my picture viewing program should use the full quality mode, and my text editor should use the fast mode. Since I'm using a tiling window manager opening a picture would then use about a second to display the image, but once it's there I could type into a window next to it at full speed. It could also allow me to highlight areas to redraw in high-quality mode if I was browsing a website for example and wanted to watch a particular image.
I wonder, is there any practical way to do 1-bit just around the cursor?
Mac OS up through System 7.5.5 works excellently on such displays, because it had to run on several models of PowerBook with black and white or grayscale passive matrix displays, which were notoriously slow. While not as good, black and white support remains decent all the way through the end of Classic Mac OS, with version 9.2.1.
For this reason, when imagining an e-ink semi-unitasker laptop such as this I imagine it being built with extremely low power hardware and running old operating systems, such as the aforementioned Classic Mac OS. I'm sure DOS/Win3.1/WinNT could stand in here too, probably more easily since 486/586/Pentium-compatible x86 CPUs are still being manufactured. One wouldn't be able to run Google Docs on such a machine, but there's a veritable cornucopia of abandonware readily available for such platforms at places like Macintosh Garden[0], and given highly specific use case of such a machine I think most people could find something that suits their needs.
e-ink displays are not even up to that standard of responsiveness yet. And, as you point out, modern software has higher expectations than the systems that kind of sucked on those old passive matrix LCDs.
A mouse pointer may itself be a bit hard to display on an E-Ink Screen. It will definitely leave a trail behind.