This is strange, I am running MSDOS 6.22 on a modern laptop, a Lenovo i7 ideapad. The machine boots directly to MSDOS. There is an external keyboard attached because lenovo swapped the right shift key and the up arrow key in some way that causes me to hit up arrow when I mean shift.
I am finding that I can press the Insert key to switch to insert mode, as soon as I hit <enter> it sets itself back to overwrite mode. I have tried swapping the external keyboard for a different one, Removing CMDEDIT and loading doskey instead, then removing DOSKEY too and going without a command editor. The problem persisted through out all of this.
I even searched the BIOS for some salient setting, no luck.
I am sure it used to work as I am remembering it - the system starts up in insert mode and pressing the insert key toggles between insert and overwrite. The setting persists thru carriage returns.