One picture sits under everything snotty does: a small child with a nose that is running, and the wet, sticky mess that comes with it. From that single picture two paths open. Keep the mess itself and you get the literal uses — a snotty nose, a snotty tissue — and then the borrowed feel of it, slippery and soft, for cold eggs or half-dried glue. Keep the child instead, chin up, answering you in two cold words, and you get the attitude use. Nothing has actually come out of anyone's nose there; what survives is the feeling of dealing with someone who is being unpleasant and thinks they are above you.
From the word’s history
Wiktionary gives the packet a simple build: snot plus the ending -y, through Old English gesnot and Middle English snot. So the word starts as nothing more than covered in snot, and every later use is that picture being carried somewhere else.
packet 裡的 Wiktionary 說法很單純：snot（鼻涕）加上 -y，往上可以追到古英文 ġesnot、中古英文 snot。所以這個字一開始就只是「沾著鼻涕的」，後來的用法都是把這個畫面搬到別的地方去。
Wiktionary entry supplied in packet (source_ref kaikki:3f6f69a190036ae4:1272581392)
登入就可以看每一支怎麼連回核心畫面。