It's the groups damage, not the tank's which matters
It's about what tank and healer combo does more damage. A paladin tank with either WHM or SCH will do more damage as a combo than a War and WHM or SCH. The ability to pull a pack in an AK or something and need zero heals until the fight is done blowing only one CD maybe two, means the healer is essentially another dps. On my WHM I can dps until the pally gets to 1.5k or so then switch out of cleric stance and top him off then go back to dpsing. Holy spam is a fantastic amount of AOE damage and I can do that knowing the paladin will survive. I can't as well with an equally geared War. They simply require more healing.
I run a geared 50 paladin and geared 50 WHM. I've healed both paladins and wars in AK's and WP's and CM's. The group does more damage and clears faster with paladin.
Look up speed clear videos in AK or WP and look for any with a War. Then look at a modest speed run with a paladin. The latter will have a faster clear time.