Well, considering I chuckled at the first 12 clips in that video you posted, I'd say I'm not sick of it at all.
I started picking up on it a few years back, and whenever I hear it, it makes me smile. Most recently in The Hobbit I noticed it, and while it did take me out of the experience for a SECOND, I quickly got back in. It's harmless, in my book. But I suppose I can understand if it really bugs you.
PS., There are Wilhelm screams in Aladdin and Beauty and the Beast?! How the hell did I never notice that?