From the perspective of someone who has studied historical martial arts a lot, the advantage of the high ground in a small scale engagement is massively overrated. As you mention in your recap Jubs, the combatant on the lower steps has a distinct advantage of reach in this particular case.
I would note however, that a tall shield would be very effective while defending a staircase. As is rightly pointed out, most staircases are too narrow for wide swings, but the notion that combatants would be taking wide swings at each other is a little disingenuous. Thrusting up into the legs and waist of man holding the stairs above would have been more common, as it is more effective, less taxing and, particularly in this case, viable in the narrow confines of the stairway.
I would suggest that the design would actually provide a small defensive advantage to the man holding the tower above, but I agree with the premise of the article that the staircase itself was not originally designed for such a purpose, and that is far more likely that defenders would hold the top of the stairs, on stable ground, and allow any attackers to make their way almost to the top of the staircase before engaging. I believe from a combat point of view, the design of the stairway would further impede a right-handed attacker, providing the defender a slight advantage in that way.
In hindsight, I should really do a video on this if I can.