I guess the 5+ on 1D6 is best for fairly arbitrary things or at least things where character's skills, attributes and abilities don't play a strong role (or if they do you bump the roll up or down to suit)Not to single him out, but I think it's best for everything, which is kind of the point of the post.
We tend to emphasize ability scores too much, in this game where originally ability scores had little effect. If you change the roll to cross a bridge to a d20 roll less than/equal to Dex, instead of a 5+ on 1d6 roll, what have you accomplished? Average Dex characters will have an increased chance of falling, Low Dex characters may have double the normal chances of falling, or more. And Dex 17 characters wind up with almost a 1 in 6 chance, with Dex 18 only marginally better. We haven't really made things much better for most characters by using a Dex check.
We could allow a +/- 1 on the "5+ on 1d6" roll for significant ability scores, which I do in some cases. But more and more, I prefer using ability scores as a trigger for when to make a roll at all. Always include an option where the players can avoid rolls, only require a roll when operating beyond that scope, for characters with an ability below a certain threshold.
It's part of my view that rolls should not be skill rolls, but situation rolls: chances that something may go wrong, or go right.