Kitzbuhel is one of famous ski places, but most of people mostly come there to show off, not to ski. Next to that, tracks, lifts etc. are normally everything but great, so most of people who actually like to ski, choose other destinations then Kitzbuhel.
But I agree... if you come from other side of ocean, it would be shame not to try it.
In my experience of Austria, things are pretty much deserted until the week of New Years, when they get massively crowded. Even Christmas is pretty quite. If the early season conditions are good, it can be a great time to go skiing.