Here's the way I'm thinking about this:

the vertical cuts do not significantly the internal structure of the onion as each individual cut I make does not entirely sever the connection between the thin vertical slices I'm making. This means that I can do a lot of these, and not worry about harming the overall structural integrity. Then I make a single horizontal cut which does harm the overall structural integrity. This is not intrinsic to the horizontal cut itself, but the fact that I have both horizontal and vertical cuts.

If I start with the horizontal cut, again I do not signficantly harm the structural integrity of the onion. However, each subsequent vertical cut I make is now going to individually compromise the integrity of the onion.

With a sufficiently sharp knife, the single horizontal cut at the end does not really pose a significant danger overall.

This all being said I almost never do the horizontal cut out of pure laziness, and instead prefer to just do angled vertical cuts analogous to the video. They're never perfect but fine enough for me...