How to classify each Job.

[Gemina]

At this point SMN is barely DoT-focused. It's more of an instant cast/oGCD-centric job that plays by a priority system. The DoTs are just a part of it. If anything, Bard has more DoT-focused mechanics but other than that, both of them don't focus on DoTs enough to even label the two as DoT-based.

SMN has one proc, Ruin IV. And it's tied to egi assaults that cause damage. A lot of SMNs ogcd madness was also reduced in the 5.1.

If anything, I would say SMN is a priority based job, but the same can be said about ALL jobs. The question then becomes what defines their priorities. SMN's DoT upkeep is without question their number one priority. This is why they also have an ogcd that applies both simultaneously, and ogcd ability that allows them to spread it to other targets, and an ability wherein its potency is increased based on both Bio and Miasma effects being active. There is also the Egi's auto attacks, which is also a DoT.

If someone was coming into FFXIV asking which job focuses heavily on DoTs, because they like those kind of jobs; I'd point them right in the direction of SMN.

Pets aren't really a thing anymore.
Eos/Selene automatically cast Embrace upon being summoned and her oGCD are triggered by the owner (scholar itself).
As for summoner, assault are on the GCD.
So yes, technically, those are pets, but from a gameplay perspective, we went to the "combine pet with owner" route so...

Also untrue. Pet AI is wonky at best, and we already know from previous expansions/patches why we would not leave pets on Sic, and always had them on Obey. There is nothing wrong with the summoner commanding the summon. SMN in particular now has two Demis, and getting these guys to come out is a pretty big part of their kit and their appeal. As for the Egis and Lily, they still have to be managed. If anything, I would just say that it is tougher to do so in 5.x because their abilities have all been taken off of the pet hotbar, but it still has to be done.