One picture holds most of this word: sticks picked up off the floor of a wood and tied into a single bundle you can lift. The word never named one stick. It named the armful once it was bound. That is where the food sense can be hung too, as a memory picture rather than a claim about history — scraps of meat pressed into one ball that holds together. The insulting use of the word is a different matter: the evidence here does not show how it arose, so it is kept apart rather than forced onto the firewood picture.
From the word’s history
From Middle English fagot, from Old French fagot, meaning a bundle of sticks; the further origin is uncertain, though a Vulgar Latin form and Ancient Greek phákelos ('bundle') are suggested.
從中古英文 fagot 來，再往前是古法文 fagot，意思就是「一綑柴」。更早的來源並不確定，有人推到通俗拉丁文與古希臘文 φάκελος（一綑），但這條線是存疑的。所以「綁成一束」這個畫面，是有文獻支持的起點。
Wiktionary, "faggot"
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