Tonight we had a fight.
5 PCs, level 4.
The wizard did the least amount of damage - I don't remember the exact value, but it was 2d6. The rogue did 32 (acted twice), the cleric about 15, the ranger 12 (both only one action) and my PC (the apothecary, with a "hulk" subclass) did 29. We were fighting a manticore, killed it when it tried to run away and my PC slammed him with an AOO (so one action, one reaction).
The manticore was doomed. It's CR2, we're level 4, and we lured it in a bad position... BUT... What did the wizard do to do 2d6 dmg? He cast phantasmal force and completely messed up the manticore. Turned an easy fight into a cakewalk.
If you don't like the manticore example, replace it with a more powerful, low intelligence monster like say a hill giant. I suspect the result would have been the same.
What I'm trying to say is that damage is necessary to win a fight (the wizard needed the rest of the party to do some clobbering), but it's often not what really achieves victory.