Yeah, you're definitely right.

The specific situation you listed is an uncommon one (with 3 pairs), but the much more common one is like this:

A3
A2

Board: AQT75

Split pot.

You certainly don't have to use a kicker from your hole cards, its just usually isn't the case that the 5th best card from both hands is on the board, unless they're both playing hands like A3 and A2.