There are many explanations as to why Adam Neely's harmonization (subjectively) works, so here's a few.
I think one of the reasons that this harmonization worked is that by starting on the F major, Neely basically just gave the next chord the function of leading into the B♭ chord. The F/A, being inverted, doesn't sound too different from the F chord, but the A in the bass would lead nicely into the B♭ of the B♭ chord. So the F/A sounds different and makes cool voice leading.
It's also harmonically similar to the iii chord (Am=A-C-E), and one common and useful resolution of the iii chord is to the IV chord, which so happens to be B♭. One could think of the first two bars as having tonic function, and Neely simply replaces what would be a (subjectively) boring I chord with what sounds like a iii chord, but the important thing is that it sets up the IV chord.
Another explanation is that there's a theory of chord substituting that says that any set of chords with the same function are easily substituted. The F/A has essentially subdominant function (even though F is a tonic chord, when inverted it doesn't feel as resolved, or "all the way home", as the F chord in root position), and so does the Gm, and both lead nicely into B♭, so they can be interchanged, giving the particular harmonization Neely derives. Note that this theory I mentioned isn't a universal, hard-and-fast rule so much as a guideline some reharmonizers subscribe to. So, both have the same function, so they're the same.
Two more bonus ideas: The melody over the second chord (A-G-F-F-G) would support most strongly the tonic function chords, since the F and A are emphasized and are part of the tonic chord, so one would probably not want to replace the Gm with anything drastically departing from the I chord; fortunately (or rather intentionally), everything I just described also describes the F/A. And of course, I'm going to get the obligatory "Music is free, and rules are meant to be broken, if it sounds good go ahead" that always shows up in the comments out of the way right here. I agree with it, but since the question isn't debating the power that music theory has to influence artistic decisions, I don't think questions like this really merit that comment.
Adam Neely is awesome.