That would not be correct. If you are unbreakable then they cannot deal 5 damage to you to break you and send you to the discard pile
Technically the damage is dealt, but the champion doesn't break because of it. The only reason for the distinction is Blind Faith can later remove the Unbreakable trait, and the Champion would then break.
As for the OP, yes it sounds like they forgot it had Unbreakable.