We are talking about the 17th most common word in current English - it is a very successful member of the language, and also a constituent of many idioms, and most of those idioms have a single meaning each. I will ignore the idioms and also the few distinct meanings which "as" can have as a noun, preposition, or adverb, because any of those are perhaps easy to distinguish grammatically and thus they are non-problematic in legal writing, and the quoted resource only objects to overuse as a conjunction.
"As" has semantic overlaps with "when", "once", "because", "since", "while" and "like", but it does not exhaust the meanings of any of those.
...since the index's inception.
You can't use "as" here.
When did you clean your teeth?
Let's simplify the playground a bit.
It is perhaps fair to forget about "since" altogether, because the contexts where "as" can be substituted for "since" do not exceed those where one could first substitute "because" for "since" and only then "as" for "because".
Analogously, we could perhaps forget also about "when", in favor of "once" and "while". It's good to realize that "as" and "when" are often used contrastively: "when" suggests a causal relationship beyond the temporal one, whereas "as" suggests that the simultaneity or the tight sequence of the two events was coincidental.
So we are left with "because", "once", "while" and "like". It is always tricky to try to identify meanings using other words (which can be polysemic themselves, or semantically overlapping with each other), but so far it seems that "as" as a conjunction includes a causal meaning and one or two temporal ones.
Now it's time to realize that the causal meaning of "as" subtly differs from that of "because". "Because" puts the focus on the cause. "As" puts the focus on the result. So while some kind of causality is still implied, "as" is the one, between those two conjunctions expressing causality, which is closer to the temporal meanings "once" or "while", and further away from temporal meaning of "before".
I am hoping that this exercise is showing that while "as" as a conjunction is undoubtedly quite broad (and thus sometimes vague), it is not scattered to several completely unrelated meanings. The single, albeit broad, underlying meaning of sameness or commonality derives from Middle English "as" meaning "so".
I assume (without any solid proof) that the 13th century "as" (meaning "so") became the second part of 15th century "whenas" (meaning roughly "when" or more exactly "once" or "while") which was later contracted to "as" (meaning "once" or "while"). However, the temporal meaning existed in Middle English already before this development, so it might be more accurate to say that "whenas" was a temporarily available synonym helping to distinguish the original pronominal/adverbial/conjunctive meaning of "so" from the nascent conjunctive meaning of "when".
Whether the causal meaning evolved directly from the underlying meaning of "sameness" independently, or from the temporal meaning, I was not able to find out.
Let me also remark that the German cognate "als" walked a similar route through Old High German "also". It has a similar range of meanings (excluding the causal meaning). I think that it means that the potential of "same" or "like" for being used as "at the same time" could well have been prehistoric.