I would think the adjective for leer would be just that, an adjective. Yet they really mean 2 very different things. There is absolutely no sexual insinuation with leery, while there is for leer.
Any idea why the adjective isn't similar to the noun or intransitive verb?
Well as a lorry driver we were taught to leer as part of the driving course! I assume the two words have different origins perhaps. Leery to me always suggests a very particular feeling that other word fits so well!
"untrusting, suspicious," 1718, originally slang, probably from dialectal lere "learning, knowledge" (see lore), or from leer (v.) in some now-obscure sense.
My personal reaction is that their first suggestion is right, and there is no connection with leer.
Build a man a fire and he's warm for a day. Set a man on fire and he's warm for the rest of his life.
The online Compact OED suggests that the two words are related. To leer originally meant to look askance or sideways. Not much of a stretch there. To be cautious or wary could be related to looking.
I suppose we will never know for sure, though the COED's entry sounds convincing. I had checked the OED, but there were lots of entries for "leer," and I must have missed that one.