To contrive is to work something out and put it together. A contrivance is what comes out of that work — a thing somebody made on purpose, for one job. The making still shows on it. Hold that picture and the other uses follow: the plan is a contrivance you cannot touch, and the criticism is what happens when the making shows too much.
From the word’s history
Wiktionary gives contrivance as contrive + -ance, with the suffix arriving through Old French -ance from Latin -antia. A suffix that turns a verb into the action or the result of it fits all three uses: the thing made, the plan made, and the making that shows.
資料說 contrivance 是 contrive 加上字尾 -ance；-ance 這一路從古法文 -ance、拉丁文 -antia 過來，作用是把動詞變成「那個動作」或「那個成果」的名詞。所以字面上就是「弄出來這件事／弄出來的東西」。三個用法剛好都落在這兩個位置上：做出來的機器、想出來的辦法，以及「做的痕跡」本身。
Wiktionary entry for contrivance, etymology tree, supplied in this batch packet as kaikki:3f6f69a190036ae4:848499342
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