The word "machaira"/"machaera" is indeed Greek in origin (μάχαιρα), and as far as I know it's still unclear what it referred to. From what I had read, it meant different things in different contexts in different periods, though it was always something edged. The Romans borrowed it from the Greeks, and though it seems also unclear what it referred to, it apparently was only used in a military context, so to refer to a weapon. I can't be absolutely certain the word "machete" derives from "machaira"/"machaera", but it's the best contestant so far. First, I literally found this etymology in a customs collection of South-West of France from the 17th century (it was used in a context where the machete was used as a weapon), and there basically is no other plausible etymology possible (I had looked in a Spanish etymology dictionary, and it was bullcrap: they claimed it was the contraction of "macho" (man/macho) and "hacheta" (hatchet)). If the English language recorded it as spelled "matchete", it's because in the South of France, the "ch" sound was often hard, and actually sounded "tch". It's not rare to find old French spelling "matchette", widely used in the 19th century.
As for the falc* radical, indeed it had a great linguistic productivity in romance languages. The word for scythe is falce in Italian and faux (old spelling faulx) in French. In France, there is a kind of stout scythe used to cut brambles and such called fauchon. Sickle it falciole in Italian and faucille in French. In Portuguese, the word for billhook is foice. There is also faca that you mention, and fação just came to mean "(large) knife", and thus is used to refer to machetes. Fação is the Portuguese sibling of the falchion/fauchon/falcione familly. So in Brazil today, machetes are called falchions. No wonder really: words and meanings used to be much more fluid even just centuries ago, since there was no centrally-organized mandatory schooling that could normalize and standardize language, and common language (not literary) was basically a juxtaposition of dialects that had evolved in a very complex fashion from what was left after the fall of the roman empire of the lingua franca that roman occupation, trade and administration, and christian religion, had gave rise to.
I'm not familiar with the book you mention, but I'm with the concept and fact. The scythe is the worst possible tool to make a weapon from, though. A scythe is really, really thin. Like, really. I've recently found a picture of a 3th century roman scythe excavated
in the South of France, and though it isn't the same fixation method as modern scythes, it's the same kind of blade: not at all fit for military use. It is known and documented that scythes were converted into makeshift weapons but... eurk... they probably didn't last long. The agricultural pole billhook on the other hand is almost a ready to use weapon. Actually the spike on the back is a very common feature on pole billhooks.
The problem is that even I, as a billhook fanatic, have extreme trouble finding any archeological documentation about medieval billhooks of any kind. Archeologists regard them as almost worthless, and they are practically treated like scrap metal. For example, in the excavation I mentioned, two billhooks and three axe heads were found, we see them on the picture, but the INRAP, the institution that did the excavation, didn't even bother to publish detailed shots, while they took and published detailed shots of actual worthless crap like metal bands for reinforcing furniture and doors. This lack of interest archeologists have towards tools is crippling research in that field, because these simple everyday objects get zero attention from authors and chroniclers, and their depiction in art (miniatures and tapestry) is extremely rare, and most of the time only pertaining with vine-growing.
So while extreme care is taken in preserving polearms, and they're displayed in museums, etc, none is taken in preserving/promoting the tools from which they come from, and there is a lack of data. Well, Italians seem to do a poor job in general at international promotion, but that machete-billhook-falchion thing is extremely interesting, rare and noticeable, and should be much better known. I don't exactly know what it is, it may or it may not be a "missing link" between billhooks and falchions, but it certainly is something worth studying.