Kinda. The system will always, always color the way the setting manifests.
That's to be expected and no one should really lose sleep over it.
And here, you jump the shark. It's directly something that designers NEED to keep in mind. They need to use it. GM's as well.
But, and I suspect this is more your point, one thing that WotC brought to the table was the idea that the game should have a flagship setting. Not just in the sense of a setting that had a few more books, but one that is incorporated into the rules.
Not quite. The rules ARE a setting. That's my point. The, as you dubbed it, "Flagship" should reflect the rules, and vice versa... and there was implicit and explicit setting material in ALL editions of D&D. In OE, it's very vague. In AD&D, there's a lot of setting built into the classes (monks and druids especially) and . 3E has a bunch of setting, too... much of it built into classes, races, magic items, and spell descriptions. Ignorable, but still present. And classes and races have strong mechanical impact, too.
I think this was one of the larger missteps by WotC. We'll never know whether I'm right because WotC also made some pretty radical changes to the rules and did a few other things that changed the landscape. There's no way to analyze what would have happened if they'd only changed the focus on setting. The flailing between settings with each edition is a symptom of this.
Well, we can draw some inferences, and both WotC and Paizo did plenty of market research before launching their games...
At the time they were prepping 3E, WotC decided they needed a flagship... but picked the wrong one. The fans generally seem to prefer the Realms. And the mechanics for the realms upped the power level.
When Paizo realized they wouldn't be able to support 4E the way they had 3E, they started their research. They, also, decided they needed a setting - Golarion is a more gonzo setting than either Greyhawk or the Realms.
Meanwhile, 4E was trying the no strong setting elements... Which, as JeffB notes, made it easier to trim and prune... but it also meant more work for the GM. 4E isn't a bad game... but between the lack of D&D feel, and the lack of a strong set of setting elements in the rules, plus no explicit favored setting...
Then, the 5E era dawned. The realms are it. No mods (just specific racial labels) for the realms. ANd 5E has set new records for sales.
Paizo isn't ditching Golarian with 2E, either.
The preponderance shows a reasonably strong correlation to good setting integration and good sales.
Hell, I picked up Pugmire for the setting.... and, largely, it's worked in through in-character narrative blocks, and "calling" descriptions. plus some introductory material...
Complex rules games all have a built in setting built into the mechanics, and this might not match the prose. People tend to complain, loudly, when they don't match.