Healers heal.

That's why a majority of their abilities have to deal with healing. They have the ability to do damage because the party questing style hasn't been in fashion for nearly a decade.

This is why tanks can do damage.

However, calling a tank a DPS without burst potential is blatantly wrong. DPS classes deal -high sustained damage per second-. This is not 'burst' damage. Burst damage is the phase where your cooldowns are popped to get the most overlapping effect from multipliers. (Black Mage Eno/LEylines/Raging, Summoner Dread/Raging/Pet buffs, Dragoons -all the cooldowns-, potions).

Tank damage is actually pretty lackluster, but they have phases of high damage output thanks to ridiculously good cooldowns with higher than average uptime. Tanks deal low sustained damage, but have high burst phases with shorter spans between them. This averages out to mediocre damage per second -but heavily favors short phase encounters-. Remember when 'omg warrior was op' in 3.0? The encounters were Ravana and Bismarck. Encounters with short windows of high vulnerability, perfectly suited to the Berserk / Fell Cleave style.