The nice thing about Desolace was the initially high numbers of demons. And then more demons. As a paladin, that was great. Exorcism was very powerful and it just generally made sense for a paladin to fight demons. I've since moved on to Feralas, with the level gap there being filled by a lot of Scholomance runs. The Scholomance runs reminded me of the joy of holy wrath stunning a big crowd of enemies. Again in Felwood, though I've only just started in there, I'm having the privilege of being a paladin fighting demons.
I also learned that Felwood has been cruelly designed to prevent miners from completing quests, as approximately every 15 yards there is a mithril vein, causing miners to wander far, far away from their quest areas until finally the bears get angry, and more importantly, level up.
I don't think it would be a very good idea to design a zone exclusively for a particular class, but designing zones with certain classes in mind could be a great deal of fun. Blizzard rejected class-specific quests, excluding rogue legendary weapons, because they were only useful 'once'. So that's out. But tuning zones for particular classes, to make their abilities more tactically useful and to fit with the general lore and motivations of a class, that would allow players a sense of the unique nature of their classes.
For paladins it's pretty obvious: demons and undead. Other classes are trickier. Hunters could have a beast-intensive zone, perhaps with some special use for traps. Priests could have a zone filled entirely with companion quests, all the companions using the standard AI of moving slowly except when pulling unneccessarily large groups of unnecessary mobs, while crying for help. Warlocks could have demons everywhere, particularly elite demons, turning it into a giant playground for them, much like Dire Maul East was, once upon a time. The rogue area would be filled with enemies in constant combat, too powerful at full health, but always beating each other down to the point that a quick cheap shot-eviscerate would kill them, and tuned to give xp balanced for the three minute cooldown on vanish.
This might mean the zone ends up easier for that particular class. So be it. As long as the rest of the game is of normal difficulty, which hopefully would not mean trivial, then having an easier zone creates a contrast, making it stand out for that class. Once again, I want to reiterate the notion that these zones are not class-exclusive, merely tuned and written with those classes in mind. They should still be fun and interesting to other classes, just more so for their particular class.
Creating entire zones would be excessive, and risks becoming far too repetitive. Limit it to particular areas, such as introductions and endings of zones. These would also make ideal locations for any class-based quests, should Blizzard decide that they were actually a lot of fun. Just yesterday I found myself arguing with someone about whether warlocks or paladins had the better mount quest. I, of course, said warlocks, because everything warlocks did was awesome, whereas paladins were just a bunch of dress-wearing sissies, with the exception of da bootiful FHPs who know what I be sayin', mon. Don't ask if ya don't know.
Musing a bit more on this, perhaps starting zones could be class zones. Death knights, regardless of faction and race, share a starting area. The story makes this work out, in addition to clever use of phasing. Expanding that concept, other classes could have a common starting area. Warriors could use a gladiatorial theme for their beginning, a story which could work for either faction and any race. Others are trickier, since, for example, priests couldn't simply have a generic priestly trial, because they don't all follow the same gods, so some amount of customization would be required. That's something I'd try to avoid.
Speaking of avoiding, the reason this could work is that it avoids the old class quest problem, which was partially that each faction, and sometimes even each race, could find itself with a particular class quest, meaning a lot of work which will rarely be seen. In contrast, if all races share a class zone, then one zone covers one-tenth of all possible characters, much like the race-specific zones. So players overall would use these zones a lot more, increasing the value of them. But at the same time, individual players may find themselves repeating starting content less, since while you might have many characters of the same race, you're less likely to have many characters of the same class. My account has a lot of trolls and humans, and on the Alliance I have almost entirely humans, but in contrast, I have a variety of classes.
Making starting areas based on class rather than race isn't likely to have any negative impacts on character perception. Ever since everyone got fear ward, have we really thought much about what race we are from day to day? Oh sure, at the starting screen we're debating the various blue shades possible to pick between a nelf and draenei, but once made, the class is what matters more. So play to that strength, developers, and design starting zones that emphasize what really matters: class.