I largely agree, and if you're more interested in helpful suggestions than in people telling you that you're wrong to feel that way, I might be able to help.
Off the top of my head, the first thing you could do is swap a few skills and features around. Allow and mandate training in one skill appropriate to the diety (so Religion + Deity Skill + 2 other Cleric Skills), such as Acrobatics for Avandra, or Arcana for Corellon (most of these should be pretty self-explanitory).
This means that even a cleric can sneak if they are the cleric of a sneaky god, which only makes sense. It's not tremendously unbalancing, since you don't really get a whole lot of extra things to do with those skills (you can't sneak attack from hiding, even if you can hide), but it gives you something distinct that you can do.
You might allow Channel Divinity to mimic one of the at-will abilities of another class, too. Say you're a cleric of Kord, so instead of "Divine Fortune" you take "Divine Might." This has an effect exactly the same as
Tide of Iron from the Fighter list, only it's an encounter power instead of an at-will power. Again, it makes your cleric very distinct, without stepping on too many toes.
Third, you may consider giving a bit of "extra incentive" to things that a cleric of Deity X are likely to do. For instance, you could allow clerics of Moradin to take Dwarf racial feats. You could let clerics of Bahamut who multiclass into Paladin gain a kicker for it (maybe they can gain the Channel Divnity feat in place of a class feature, maybe just give them a +2 bonus with a certain class feature). Clerics who pick up proficiency in a favored weapon might gain an extra +1 proficiency bonus, or they might be able to do radiant damage once per encounter with that favored weapon.
Alternately, you could go the other way around. A fighter who picks up cleric levels might automatically gain the Channel Divnity for Kord. You might also consider modeling it off of warlock paths -- Deity Paths!
All of those are pretty "quick and dirty" solutions. Something more in-deapth would mean, say, creating paragon paths or alternate class powers at several (if not all) levels that evoke more specific deities in the way that warlocks can evoke specific paths.
None of this should be too tremendously burdensome in isolation, but there are two things you're going to want to watch out for.
The first is Niche Protection. 4e has a very precarious balance between the four roles. There is room for flexibility, but it's important that your cleric decide exactly what role he fills. Does he want to be mostly a leader with some tricksy rogue elements? Does he want to be mostly a tricksy rogue who can inspire others to be tricksy? You can't have it both ways in 4e, so you've gotta do one or the other.
The second is Accidental Suck. This is related to the above problem. You don't want to end up with a cleric who can't heal because he's traded out all his healing powers for fighter powers, but he can't fight that well, either, so what, exactly, are all those fighter powers for?
Clerics in 3e don't really suffer from the latter problem because they pile onto the first problem pretty avidly -- they can fill any role, and they won't suck at any of them (a bit of an exaggeration, but only a bit

). I think getting rid of that problem is part of what drove clerics in 4e to embrace the "Blast 'em With Light" archetype.