Here is my take on things. In this post, I will first explain why I prefer classless systems as a general rule and why all arguments against it that are not exclusively about ease of use are null and void. Since I don't see a revolution coming in the way DA does things, I will then make a few suggestions about how the existing system can be tweaked to allow for more character diversity.
Part I: the three basic arguments for a classless system
1. There shouldn't be any restrictions on which skills characters can combine which are not based on the natural laws (rather than the culture) of the game world.
The thing is, classes are stereotypes. They're based on lumping skills together that we traditionally see connected because of our cultural history. Some game worlds make their own classes based on their own cultural predispositions, and that's a bit better but there is still no in-world reason at all why individual characters have to adhere to class restrictions if they're just based on cultural predispositions.
An example: If I want, say, a warrior who can cast one or two spells to boost his combat abilities, I can construct a history for him in the DA universe that does not break any lore. Maybe he's a Circle refugee who didn't want the "soft" life of a mage, fled and didn't care to continue his magical training, learning swordwork instead. As long as there isn't any natural law that prevents him from combining certain skills (possible example: being a blood mage precludes becoming a spirit healer and vice versa), any skill should be open for him to learn. Maybe he'll never become as good at a skill learned late as a specialist, but there can be several reasons why I'd want it nonetheless. Maybe I'm content with opening just 75% of all locks instead of all of them, or with casting the one or two spells that give me an edge in physical combat rather than changing my style of combat completely.
2. Anything that can be done with a class system can also be done with a classless system, but not vice versa.
Example: In a classless systems, there can be traits which can only be selected at character creation because of underlying laws of the game world. "Mageborn" would be one such trait. I make a distinction between "mageborn" and "mage" because even if you're mageborn, there is no law of the game world that says you actually need to get mage training and "become a mage" eventually. If your game world has certain rules about magic, a classless system can easily be adapted to incorporate them.
There can also be preset trait combinations, usually called "templates" (as suggested in the above post by Baron Samedi), which recreate the skill combinations that make up the classes in a class-based system. Those make character selection fast and easy for new players, those satisfied with the stereotypes or those who don't want to bother with the details of a complicated character generation system. As an added advantage, the restrictions are not carried over (unless a natural law of the fictional world says so) so these templates can still be tweaked as the player desires.
However, if you start with a class system, any character concept that softens the boundaries between classes, regardless of in-world plausibility, needs either an extra class or rules of exception, which makes the rules cumbersome and undermines the class distinction the system was created to sustain in the first place.
3. Classless systems can actually create more distinctive characters, it's just the responsibility of the player to do so.
Consider the above example of a character who breaks the mould. Classes are stereotypes, and as a rule, characters become more interesting if they break stereotypes. The disctinction of a class system is that it forces players to adhere to stereotypes, it forces a particular kind of distinctiveness while preventing other kinds.
About the argument that you'll always create "the" most powerful character based on "the" skills most powerful in the context of the story told and its gameplay elements:
Balancing in classless systems is done based on a points system. Those are indeed somewhat harder to balance than class-based systems but they do actually balance things (consider the way Kingdoms of Amalur did this). That you can combine the most powerful skills is, most of the time, not true, as having, say, sword and magic proficiency means you're not as good in either as a specialist. That this kind of balancing works you can see in the Fallout games, which haven't been accused of having a uber-powerful character template though certain traits have been accused of being overpowered (that always happens).
More to the point, though is this: if such an "abuse" of the system is really possible (I would deny this is an abuse since the uber-powerful template is just as legitimate as others), it's up to you as the player to avoid using it, and the system gives you the tools to avoid it. It's all your resposibility because class systems are about restrictions but classless systems about options. I say we players don't need our hands held in order to create interesting and diverse characters, because most of us actually want those.
Part II: What can be done with the existing system
DAO points the way. Make more character abilities available cross-class unless Thedas' natural laws prevent it. To balance that, those abilities can be attribute-based (like lockpicking already is) so that those classes which have an ability's base attribute as a primary attribute benefit more from acquiring the ability than others. Example: if lockpicking is generally available and based on Cunning, rogues will still tend to be better at it since they tend to develop Cunning more than the other classes. It may cost a mage or warrior more to develop a useful lockpicking skill, but if that's part of a player's character concept and they're willing to sacrifice other benefits for it the system should not aim to prevent it. The resulting character becomes more distinctive for it, and isn't that what we all want?
Also, do *not* add classes. More classes means more enforced stereotyping, and more restrictions.
And remove *all* weapon restrictions except non-mages being unable to shoot magic projectiles with a staff. Instead, give warriors skills that make them better at certain things so that while, say 2h-weapons-skill tends to be more useful to a warrior, others aren't prevented completely from using such weapons. Treat staffs as a 2h-weapon and give mages special abilities based on a staff's magical attributes rather than their physical ones, which means warriors could bludgeon harder with a staff while mages, for instance, could add elemental effects to their melee hits. In other words, make class distinctiveness a part of the characters rather than part of the equipment.