From the skill lines they've released so far, we can see that self heals are pretty darn limited. With the exception of the Templar, self heals are only minor things that come from passives.
The fact that you're going to need dedicated healers is what keeps the trinity alive.
Also, just because you could potentially do everything doesn't mean you can do everything at once. Its one of the great things about the 5-slot bar. I can unlock many skills, but only have a few of them available at anyone time. I can use any armor, but only one at a time (or I can get lesser but more varied boons by mixing.) I can use any weapon, but only one (or two with weapon swap) at a time.
That means that I'm still going to have to spec into a particular role if I want to be effective.
Jack-of-all-trades builds often svck, even in TES games. Sure, in Skyrim you could be a sword wielding mage who practices every kind of magic, sneaks around, and uses a bow. But if you spread your perks around that much, you'd get mauled on the highest difficulty settings. You best option is to specialize at least to some degree.
In ESO, I can probably be a DPS/healer. But I'm not going to be as good at DPS as a full DPS and not going to be as good at healing as a full healer.
That's what the classes give us. Unique strategic choices with trade-offs.