It sounds like D&D is the wrong system for you, but the right system for your players. If you don't want to switch groups, you'll have to find a modus vivendi (which can also be letting someone else DM). Mechanics alone won't solve the problem of incompatible playing styles.
IMHO the main appeal of D&D is "hacking" the system to build the character or concept you want in a mechanically sound way. It has rather inflexible rules, but lots of them (sometimes contradictory, sometimes multiple solutions for the same problem that only differ in technicalities). This puts it pretty far in the "gamist" corner of systems ("narrativism" is IMHO an attribute that can't be applied to systems, only to play styles - you can have a narrativist approach to D&D, but a pure gamist won't find it enjoyable - and a pure simulationist won't find D&D enjoyable anyway).
Also, don't think too much about "your story", as this tends to lead to railroading. And while gamists usually don't care too much about railroading in a purely story-wise sense, they WILL be upset once you limit their mechanical choices without good reason or present them with "unfair" encounters.