AbdulAlhazred
Legend
What if I want to play a game style in which those out of combat details matter?
I suppose that's one of the places where I don't have an answer. DM Fiat works well in a lot of cases, but there are also a lot of game styles I like to explore in which that doesn't work well.
What are some of the solutions you would suggest?
note: Once again, I am not defending 3rd Edition. I'm instead curious how someone might further tweak 4th Edition to better suit some of the things I want to do.
Depends on what things you want to do. 4e has a number of rather open-ended aspects. You've got backgrounds, which effectively you can use to cover almost anything. There are published backgrounds that do all sorts of things, and just the standard skill or language thing is pretty useful by itself for adding some little element to round out your character concept.
So for instance if I wanted to be a character that plays the lute then I'll just say to the DM, "Hey, I'm taking the Professional Musician background element (or whatever it is called, I'm sure there is one)" Done. When you want to influence someone you can describe it as using your instrument to play music if the situation is at all appropriate to that. This can be used in an SC or whatever. You can apply it to any other skill for that matter where you can make a logical case for it. For that matter if you say have the Healing skill you could play soothing music. It doesn't have to do anything, though if I really wanted to make that a really important part of my character I can say ask for a +2 to checks where I'm able to play. It is a method-oriented system that isn't attempting to codify what you know, but to describe how you operate.
There are of course many other things you can do. Pick a theme that gives you some specific capability, use skill powers to become particularly good at specific things. There are a pretty decent variety of other utility powers that are useful here as well, and various racial feats and such. Then there are just feats in general. There are actually quite a lot of feats that are useful to do specific things.
What is nice is none of those options says "you have to know this to do X, nobody else can". Niche skills tended to do the opposite whereas a power that gives you a nice bonus to do something or a reroll has a different message. There are always the general skills you can use all the time. Martial practices are a bit of an oddity in that they work more like skills than anything else, letting you do something very specific whenever you want. I'm not sure they were really a great idea, but they certainly are a good resource for picking up interesting abilities that don't compete with anything else.
I guess you could also count boons, though technically they fall under 'treasure' they are really a type of freeform training.
Then of course there is the sometimes maligned but powerful ability to just fluff things. The level 30 invoker that can't swim just walks across ordinary bodies of water (with at least a +15 Athletics he may be weak in that skill, but he's able to do things normal level 1 people can't). The wizard mutters some minor incantation and picks a lock with thievery. Albeit the rogue would do it with much greater facility, but he can do it. These are all skill checks but they need not imply that the character can do something that for RP reasons or just lack of training the PC "can't do".
I'm really not sure what other things you were asking for specifically.