You can picture a crutch as a stick that forks at the top, so that a body can rest in the fork and push its weight down through the wood to the ground. Two things sit inside that picture: the weight being handed over, and the fork that receives it. Follow the weight and you get the figurative crutch — the thing you hand your own load to. Follow the fork and you get the British use for the place where the legs meet.
From the word’s history
Wiktionary traces crutch to Old English cryċċ (“crutch, staff”), from a Proto-Indo-European root meaning “to turn, bend”, so the oldest idea in the word is a stick that bends or forks at the top.
Wiktionary 把 crutch 追到古英語 cryċċ（拐杖、杖），再往上是一個「轉、彎」的古老字根。所以這個字裡最老的想法，就是一根上端彎出去、分出叉口的棍子。這也解釋了為什麼「叉口」這一層在英式用法裡還活著。
Wiktionary (kaikki snapshot 3f6f69a190036ae4, source_ref kaikki:3f6f69a190036ae4:1180404464): “From Middle English crucche, from Old English cryċċ (“crutch, staff”), from Proto-West Germanic *krukkju … from Proto-Indo-European *ger- (“to turn, bend”).”
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