Arguing about which is more important is arguing whether skin or organs are most important. Without adhering to at least some of the core rules you end up with over/underpowered characters or players being treated unfairly. The story will actually end up suffering if there isn’t some mechanical framework. This is fine if you are writing a novel by yourself, but once you add multiple people into it the story will bog down if there isn’t a way to keep it moving. Often rulesets have been built to speed along the kind of play the game was made for.
Conversely, if you have too many, or wrong for the story, rules then you just end up consulting charts and hunting minute rules down in the rulebook.Exciting for math class…not too much beyond it.
In the case of D&D, its current rules are designed to follow a certain flow. Fight- level, fight-level, fight-level. Each time the challenges get bigger in scope and power leading to a (hopefully) climatic final battle. Pushing against that flow results in dull play.
This also means you should find the rules that support the kind of play you find interesting. Looser rules? light rules? Rules that reward narrative play? denser rules? super simulation? Hyperrealism? All a matter of personal choice.
I want to run weird fantasy…but not too weird. I don’t want my players being sentient eyeballs and shit. Fighting sentient eyeballs though…damn, gonna have to work on that. I would not choose D&D because I feel like I want to escape level based advancement. I also like a more concise wound system for this.
The bottom line here is that if you don’t pay attention to how they interact then your game will suffer.
For example running Star Trek using Shadowrun rules…bad. Judge Dredd using Storyteller…good. Why? Because Trek stories are about discovery and exploration while Shadowrun is about combat. oil and water there. Dredd, surprisingly, is about investigation and themes of law versus morality which meshes well with the ST system.