I don't think every healer strictly needs a DoT. They could achieve the gameplay of maintaining something in different ways.

WHM could apply a debuff that slowly boosts the potency of a spell over time, so you maintain the debuff and hit the spell at max potency to not overcap.

SCH can be the healer that keeps and manages actual DoTs.

AST could have a damage compile ability to keep up or an ability that grows stronger over time but needs to be detonated manually by Combust.

SGE also doesn't really need a DoT, I feel like direct damage and juggling short CD damage buttons should be their specialty, maybe they could apply short burn DoTs from Phlegma and Pneuma.