It's not arrogant if it's the truth. And considering how easily he defeated Kakashi and Sasuke with his Sharingan, he further proves that point. Itachi mentions to Kisame that he didn't want to cause a commotion which would attract attention. If you want to win a battle quickly, you're not going to use mediocre techniques which the enemy is capable of countering or defeating, you're going to use something that an opponent has little or no chance against.

And for you statement concerning Kakashi copying Rock Lee opening the gates, this is just pure speculation. Considering Kakashi's knowledge of the 8 Gates, he probably already knew how to use them prior to the Lee and Gaara match. It seems the only reason he unveiled the Sharingan was to keep up with Lee's speed.