Not to sound contrarian, but I don't know if another toggle setting is the way to solve this issue. I think this touches on a few odd or incongruous design choices in the MIDI editor.
I just did some quick tests on Reaper 5.70 x64 on Windows 10 with the following settings enabled. I wish I could have been a little more thorough, but with my workflow I don't typically have continuous scrolling with playback enabled.
[x] Main window -> Options -> Continuous scrolling
[x] Main window -> Options -> Automatically scroll view during playback
[x] MIDI Editor -> View -> Piano Roll timebase -> Project Beats
Things that I noticed and was/wasn't able to replicate:
1st playback test: The arrange view scrolled smoothly and the MIDI editor scrolled in chunks equal to the current horizontal viewing window of the MIDI Editor.
1st recording test (overdub mode): The arrange view scrolled but the MIDI editor would not (as it was in your video).
2nd recording test (overdub mode): The arrange view scrolled and the MIDI editor scrolled in chunks equal to the current horizontal viewing window of the MIDI Editor. Strangely, I was not able to figure out what changed between the 1st and 2nd test to make this happen.
So essentially, sometimes I could replicate your issue of Project beats timebase not scrolling and sometimes I couldn't. Then I did some quick tests with the following setting changed:
[x] MIDI Editor -> View -> Piano Roll timebase -> Project synced
The issue you brought up about the view jumping around seemed to depend on where in the MIDI Editor the edit cursor was positioned when I pressed record. If it was further left in the window, the view would reposition multiple times as the playback cursor scrolled left to right. If it was further right in the window, this behavior would be smoother and less erratic.
Seems to me like something that could use more testing, if any other users want to help out. Didn't you have a thread before where you had an issue with the MIDI Editor sometimes scrolling and sometimes not scrolling horizontally? After my testing, I wonder if these are somewhat related. I definitely think all these behaviors are odd but I also think they're symptoms of a design issue. I'm not entirely sure adding a checkbox to the already huge list of options will do it this time.
Just for the record, someone sort of brought this up 3 years ago, but I don't think they were super clear with what they were looking for so it didn't really get implemented.
https://forum.cockos.com/showthread.php?t=135768