I have 17 empty cells in my final grid after following the end game instruction - and still have some non words across. I chose to take “only the letter checked by a down entry must be entered” to mean all unchecked letters are left blank. Indeed at this stage there is real uncertainty about what these letters blank cells could contain.
After completing the end game that uncertainty is removed.
You could argue at that point only one identified cell in each across entry requires 2 letters and the instruction could be read to say only the letter IN THAT CELL corresponding to a down entry must be entered.
I chose to leave all the unchecked cells blank after the end game but this means you do not need to complete all the across clues for submission which seems a bit unsatisfactory.