> it occurs to me that his issue with the reversal > of quantifiers in relation to directed set structure > is the key.
You are in error. The argument is much more complicated than primitive quaantifyer reversal.
The argument is, unfamiliar as it is: d is nothing but every FIS d_1, ..., d_n. And it is obvious that every FIS is a line. This statement can only be checked up to every finite n. Therefore d is not ouside of every line.