Hm, I had the opposite problem. I found the class descriptions pretty flavorful, but the mechanical bits at the end of that section, just before the power catalogue, were mushy and ill-defined as far as I was concerned. As has been pointed out before, though, so much of that stuff is all about combat. Blades in the Dark, for all its faults in organization, really won me over with its playbooks and special abilities that could be about so many more things than whacking somebody.
That's a good point: 4e classes are basically all about combat, which is probably also why they are so closely tied to roles who are also all about how you play in combat. So if you expect your class to tell you non-combat things about your character, 4e will disappoint you. That doesn't mean that the
game is all about combat, just that the non-combat mechanical parts primarily lie outside of your class (mainly skills and rituals). I don't know if the later addition of themes did anything to change that, because by that time my game group had decided that 4e wasn't for us.
One issue here is that most D&D settings give classes diegetic representation within the setting. Being a cleric doesn't just mean that you channel divine power in order to guide and support your allies, it usually brings with it a role within an organized church of some sort, and the important people of that church will predominantly be clerics and maybe paladins. Being a wizard doesn't just mean you cast spells that either hurt multiple opponents or confound and harry them in various ways, but also that you're part of a tradition of scholarship that dates back to the ancients, and that you're a frickin' nerd. Being a warlock doesn't just mean that you use magic to inflict horrible damage and curse your enemies, it also means that you have struck a bargain with some form of otherworldly being in exchange for power. The martial classes have somewhat less of this, being more of a blank slate where you can write your own story, but they still have some of it.
One of 4e's mistakes here was probably to (a) hardcode this connection between story and role, and (b) not do all that much with the story part. One way of fixing this would have been to lean harder on the story part and give that some mechanical heft, while at the same time providing different in-class methods for fulfilling different roles. If going down that path I would take care not to add story stuff to the sub-classes – it's cool if you have some "hype man" bards who uplift and support their allies and other "dis track" bards who bring down and befuddle their foes, but they should not be in different "bardic colleges" or whatnot. They should just be different approaches to bardery. Another way of fixing it would be to separate story and role entirely, but I think that would be really difficult to do in a satisfying way.
Edit: Come to think of it, I think 5e tried some of that last thing with backgrounds. The idea being that what made you part of organized religion was the Acolyte background and not the cleric class, and your Urchin Cleric might just have been some dude that got randomly blessed by Olladra. But that doesn't really
feel right – if you have divine powers, shouldn't you at least need to have some form or relationship with the appropriate church? Even if you aren't into the hierarchy, it seems you're either going to have to join or explicitly reject it – either way, it remains a relevant thing.