Could someone please elaborate on the proof of Corollary 3.2.?
Why does one need a total order *on $F$* if one is just interested in well-ordering each $F_x$ (in order to choose one element)? Also, a well-ordering on a finite set shouldn't be hard to find, the problem probably is to find a well-ordering on each of the possibly infinite many finite sets at once.
The point is that if you have a family of finite sets whose union can be linearly ordered, then you can uniformly define "the minimum" from each of those sets, and thus have a section. And this is the catch: moving from "each set can be well-ordered" or "all sets can be well-ordered at the same time".
For example, even if $\Bbb R$ cannot be well-ordered, its finite subsets can be uniformly well-ordered, by taking the linear ordering they inherit from $\Bbb R$.