I liked your first post in this thread, but couldn't disagree more with this one.
1) Not all SCH's liked the DoT focus. /raises hand
2) SOME players appreciate WAR, many do not. Many Tanks want to play a tank and not a healer. Many Healers want to play a healer and not a tank or a DPS. Making Healer Jobs into DPSers and telling them to go play WAR if they want to heal seems bad on all counts.
3) RaionKansen is right that "more dots" isn't the answer. At best, it papers over the problem of Healers feeling unnecessary in even current content. It doesn't actually fix the problem, it just lets them be even worse and more braindead (but SLIGHTLY less braindead) DPS Jobs. Say SCH got a 15 and 18 second DoT, one with a cast time, and an AF spender that did more damage based on DoTs. Is that really going to be more compelling for long? You're not hitting Broil as much...but you're not hitting it THAT much less often. If it makes up 80% of your casts now, it'll still make up 70% of your casts after the change, and it doesn't take a lot of brainpower to go "look under boss health bar, make sure it has my three DoTs on it before hitting Bane, repeat until AF all gone".
...at WORST, it just makes the role into worse DPS and alienates people that genuinely like healing into quitting the game or leaving those classes. If someone likes healing - and I mean "filling up people's HP bars - and not DPS rotations, then all "more dots" will do is encourage them to play RDM or possibly SMN or DNC instead, maybe WAR or PLD, or quit the game, neither of which is great for the game. Groups will start dropping Healers, as why take a Green Job that has the rotation of a Tank but does even less damage and brings nothing useful to the table when you could just bring an extra WAR and an extra DPS instead and clear the encounter just fine, and faster?
I do think there's some design space in making the Healer JOBS more distinct - SCH having DoTs and SGE having a combo rotation, for example, leaving AST and WHM basically how they are now; that might work - but "more dots" won't actually fix the problem, and even giving just SCH more and SGE a combo rotation won't really fix anything, it'll just steal some of the hyperbole from the people who want to be "dps-lite Support" and want to coopt the Healer role into being that when the Tanks are already that and they don't want to play those because they don't want "the responsibility" or...something.
4) If you're going to remove Healing from the game, you'd be better off just deleting the four Jobs instead of trying to change them into "dps-lite Support". You'd also need to make Jobs for the people that play them now that they might vaguely enjoy, meaning you'd need some DPS Jobs that have an automatic DPS rotation for the people that don't really care for DPS rotations, and that would just be stupid.