I believe the idea of 'The Healer must DPS' comes from the fact that noone in the party wants to see the healer just standing around for 5-12 seconds.

But the issue is because its so easy to top-off tanks and people with off-gcd heals + a single cast, there's usually nothing TO heal. That or the heals are just so bloody strong that a single cast restores most if not all of the tanks health. So if there's nothing to heal... you either stand or DPS.

And people hate to see others just standing for extended periods of time.