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If your pump moves water from the ice box to the sink, then it's not connected to a through-hull and can't sink the boat. I would like to know why your sink drains so slowly. On my boat, the galley sink is connected to a through-hull and drains very fast. Perhaps some chasing of plumbing lines is in order . . .
I was a little stumped too - my H36 sink drain is an above-the-waterline thru hull. It was on my C27 as well. I can't speak for Kappykaplan's setup, but I think that's typical. So the drain thru hull SHOULDN'T be the reason for outside water coming in (but it could be impeded, slowing the quick draining of the sink, as Kappy mentioned).
HOWEVER...
I think the confusion is arising because Jbertelli is referring to an above-the-waterline drain which is draining slowly, and Kappykaplan is using the word "drain" to refer to winterizing the seacock thru which the footpump gets its outside water, and allowing that seacock to "drain" when on the hard so as not to get freeze damage which compromises the seacock thru hull (and not the drain thru hull which is above water). And absolutely - if the seacock or the plumbing attached to it (if the seacock is open) is damaged it can definitely result in outside water coming in, and potentially sinking the boat.
Hopefully I haven't confused things further.