As much as I love these posters above for saying all these great things about old school games, I feel obliged to give a few opinions on how old games were originally designed and how they can't quite do
everything. At least without understanding what needs to be done to them first. You don't have to play them any particular way because of the original authors' design, just keep in mind what they were originally understood as.
And that would be from 70's wargaming knowledge of the U.S. military's role playing exercises from that time. D&D was something different. It wasn't a wargame hobby simulation, so I believe they looked around for the next likeliest answer: role playing exercises. Unsurprisingly, the military picked D&D's kind of simulation too (calling them map exercises (
link)). The point here I'm trying to convey is these exercises were "be the best soldier you could be" role playing, not explore your character/ personality role playing. In D&D and other games like it the objective is to be the best fighter, or wizard, or thief, or whatever, not "let's explore this theatrical role or what it means to be this fictional character". (Think social roles not theatre roles) Anyway, get as much XP as possible and you're probably doing pretty well at what the games were designed for.
If you want to do character exploration as you say in your OP, that can still be done, but it requires player or maybe group initiative to do so. The game design isn't built to specifically enable that as most Indie games are.
And if you want to tell a story too, that shouldn't be hard either. You've probably heard of Particapationalism. Adventure module one-shots linked together do pretty well too like Adventure Paths or G1-3. You just want to be up front with the objective each time: "this adventure is about saving the town" or "clearing out this old castle" or "Capture Red Team from high ground with no losses on either side, all prisoners". (That last is just so you see where this is coming from). I say this as the setting /situation / game objective here really is the plot. Don't give them an objective and you'll be (practically) plot-less.
That said, you should absolutely look at other things beyond what I think others called Neo-old School. Stuff like megadungeons and being hard on early PCs and whatever else. Just remember that if you want to design for a different kinds of objectives, they should at least somehow test the Players' abilities to succeed in their class. Otherwise you'll lose much of the rule support. I.E. A combat and magic system, but no farming system, kingdom rulership system, mercantile system, basket weaving system, glass blowing system, etc.

At least not in D&D out of the box. Those aren't the roles supported.