Last postseason a NBA basketball game took
33 minutes of real time for 90 seconds of gameplay, because the refs kept reviewing each call and debating what was correct or not. IMO, it's not using the rules that is them "getting in the way," it's when the adjudication takes up so much more time than the actual playing of the game.
I think that's a very good way of putting it.
Yes, I know that instant replay is a MUCH more fair an accurate way to judge sports. I know. I totally understand the rationale.
But, it does MASSIVELY slow down play. I love NFL. And, I think the instant replay rules are pretty good. But, good grief does it get awfully tedious sometimes. And, I would never want an instant replay rule in, say, my backyard game of touch football. The stakes just aren't the same.
Look at 3e's grapple rules. There's a perfect example of rules getting in the way. These were baroque rules that were not easy to adjudicate and came up infrequently enough that not many people had the rules memorized, but, often enough that it was a problem. And it would grind the game to a halt. It certainly didn't help that every monster and its mother had improved grab at higher levels.
So, roll forward to 5e. Grapple rules are now a paragraph long. Simple, clean, elegant, easy to use and intuitive. That's, I think, what people mean when they say the rules should "get out of the way". Another example would be the evolution of initiative in D&D. The rules for initiative used to take up more than a page. The ADDICT document, while it's probably longer than it really needs to be, is still EIGHT PAGES long. Now, initiative takes up what, a paragraph? And, it's a standardized die roll that you do for everything else.
Again, I think when people say that the rules should get out of the way, what they mean is they want the rules to be intuitive enough that you don't have to grind the game to a screeching halt for half an hour every time this particular bit comes up.