Thought I once heard fag used in British English (perhaps obsolete) referring to someone who was one's chum in one of those all-male British "Public Schools" (= "Private Schools"). And I didn't get the impression at the time I saw it that there was any sexual connotation.
Also, I know I've seen faggots used to describe a bunch of sticks or kindling or similar.
I wonder if these usages led indirectly to the common meanings for these words as we know them today? I can see how a word for a close friend in an all-male school might eventually morph into some other meaning. But as far as the sticks go ..... no! I won't go there!
You are right. In those Brit boarding schools, underclassmen had to "fag" for a particular upperclassman. Run errands for him, etc.
Some definitions of "faggot":
"faggot" is a term which over centuries has had a number of legitimate meanings, as chronicled by dictionaries and wordsmiths.
In medieval times, faggot referred to a bundle of sticks or firewood. In the 18th century the term described an underclassman's status in relation to an upper classman at an English boarding school. In the 1800's a shrewish, ill- tempered woman could be called a faggot. And in contemporary British slang, a " fag " is a cigarette.
We need not explore further morphings & mergings, into the full blown phase, "flaming faggot".