Does Cut the Rope on iPad support Game Center at all?
As of now with version 1.8 of Cut the Rope HD on iPad, when the app starts, it says "Welcome back, mike", which is the standard greeting of Game Center.

But there seems to be no where to be found ...
https://cdn.sstatic.net/Sites/gaming/Img/favicon.ico?v=103dcfc09166
Arqade
Cut the Rope HD for iPad does have Game Center: you noticed the Game Center welcome notice, which indicates the app has implemented the Game Center API, but you can also confirm an app has Game Center by looking for the Game Center icon on the app's App Store page:

As of Cut the Rope HD 2.0 (released October 25, 2012 in the US), Game Center is integrated directly into the game and you can find things like achievements in the options menu, just like Cut the Rope Experiements.

Prior to the 2.0 release, being an early Chillingo game, there's no way to access Game Center features in game. The Crystal you noticed is for Chillingo's version of Game Center, aptly called "Crystal".

Instead, use the dedicated Game Center app, select the Games tab, and find Cut the Rope HD there: you'll find both leaderboards and achievements.