I like tanking. For me, few things are more enjoyable than watching the entire enemy focus you in utter desperation to kill you and fail.
A few champions are really good at tanking and dealing damage at the same time. You just need to know how to properly build them, and most people don't know how, because they associate damage with stat numbers too much. My favorite weapon to build for tanks (like Braum, Volibear, Nautilus and Thresh) is Wit's End. It deals 42 magic damage and provides a decent amount of magic resist and steals more from the enemy (a stat most neglect to build). Tanks that specialize in dealing magic damage through their skills (Rammus, Sejuani, Zac, Nautilus, Amumu) benefit from building LIandri's Torment as their primary damage item (though Thornmail works just as well for Rammus). Tanks that have a lot of basic attack focus (Nautilus, Volibear, Braum, Vi, Udyr, Trundle) should focus on upping their DPS and maximizing what they can do with on-hit effects (Sheen items, Wit's End, Black Cleaver, Frozen Mallet).
In the end, though, yes, they are meant to be team players, and allies should know when you're ready to charge the enemy.