Wow you are soo right-- cock has so so many meanings-- in today's NYtimes crossword-- Clue-- Commode part--
answer--(Ball)cock-- and i have heard cock -- in a similar manner for a faucet-- turn the cock-- to open the valve..

and the bird-- (which gives rise to other meanings!) as well as cock up...

I can only think (no OED handy) it must be a fairly old word-- old words keep picking up new meanings-- and sometimes forgets to lose their old ones... what do faucets (or spigots) have in common with roosters? or how is it that these two radically different things came to have the same name?

or is there a third meaning to cock-- that hold the key to the link? (as in "ride a cock horse to banberry cross..." what is a cock horse? -- (well yes, i know its a broomstick with a "horse head and reins" that children ride... but why a cock horse? I can't remember what we call them-- (a pony stick?)


my other obsession