I can't reproduce your google sheets bug, it always highlights the next cell for me (not in edit mode) regardless of whether I input data into the formula bar or the cell. Are you sure it's not an extension or something?
Oh God, after reading your post, it suddenly struck me that maybe I should try using a different Google account, even though I don't really have anything special in my main account.
I registered a new account, and voilà, it suddenly worked!
I compared all the options and couldn't find anything that would cause this until I discovered that my main account's Tools > Accessibility Settings had "Turn on screen reader support" enabled, for which I have no idea why.
I turned it back off, and the issue immediately disappeared.
To be fair, I think enabling this accessibility option shouldn't cause such a bug; but for now, I can finally use it again.
I can reproduce it with Chrome, Firefox, both with no extension.
On three computers.
In case it was not clear, here is the full STR.
1. Click on a cell.
2. Click on the formula bar.
3. Type something, Press enter (you can even directly press Enter).
Expected: the next cell is highlighted but not in edit mode. So you can use arrow keys to move highlight around.
Observed: the next cell is in edit mode; you cannot move arrow around.
To make it worse, sometimes (but not always), if you press enter AGAIN, it would go to the next (the third one) and STILL in edit mode; you need to press enter AGAIN, which makes the highlight move to the fourth cell, only then it properly exists edit mode.
The bug is also reproducible with Esc (cancel change):
1. Click on a cell; 2. Click on the formula bar; 3. Directly press Esc.
Expected: exit edit mode
Observed: you are still in edit mode. You have to press Esc again.
I registered a new account, and voilà, it suddenly worked!
I compared all the options and couldn't find anything that would cause this until I discovered that my main account's Tools > Accessibility Settings had "Turn on screen reader support" enabled, for which I have no idea why.
I turned it back off, and the issue immediately disappeared.
To be fair, I think enabling this accessibility option shouldn't cause such a bug; but for now, I can finally use it again.
Again, thank you for taking the time to test it!