A little more about sch: it has the ability to, as Szabo noted, to create skillchains purely off of spells. This means that a sch can make SCs, and blms (and the sch) can nuke for a manaburst all without requiring melee to be involved. For some content that has particularly brutal aoe or aura, this is great, because it means only the tank is close to the mob. Sch also had a period where it was (ab)used for its ability to rapidly stun. That isn't as common anymore, as a lot of newer content is immune to stun, but it was a thing for a while. If you're worried about getting stuck healing on sch... well, I often joke that I never did the quest to unlock light arts for my sch.
Blu and drk can be great DDs, and a lot of fun to play, but understand that there will be a large time investment getting them geared up. In contrast, sch and cor can perform some basic functions with moderate gear and, in the case of sch, 550 JP. This isn't to say that they don't benefit from good gear--a well geared cor, for example, can be an incredible DD--but they do have a slightly lower bar for being able to contribute to a group.
"We are Dragoons, and to make short work of foppish cavalry is our bread and butter."
-Bob Shaftoe, The System of the World