Hold one picture for this whole word: a body going in through a gap that looks too narrow, following the curve of the opening, touching neither side, making no sound. Nothing is pushed. Nothing is announced. The thing is simply outside, and then it is inside, and you cannot name the moment it crossed. Everything else this verb does is that same entry performed somewhere other than a doorway — into a person's trust, or into a conversation.
From the word’s history
Wiktionary gives insinuate as borrowed from Latin īnsinuātus, from īnsinuō 'to push in, creep in, steal in', formed from in- ('in') plus sinus ('a winding, bend, bay, fold, bosom'). The modern picture of following a curve inwards sits directly on that root.
Wiktionary 記載 insinuate 借自拉丁文 īnsinuātus，動詞 īnsinuō 的意思是「推進去、爬進去、偷偷進去」，由 in-（進入）加 sinus（彎曲、褶皺、凹處、懷抱）組成。今天「順著一道彎鑽進去」的畫面，正好貼在這個字根上。
Wiktionary entry 'insinuate' (verb), source_ref kaikki:3f6f69a190036ae4:786117821:19207, supplied verbatim in the evidence packet
登入就可以看每一支怎麼連回核心畫面。