From what I've witnessed consulting, this varies greatly by company size. I once heard a Vice President of major bank explain it like this: "Look, beyond a certain size, almost nothing you can do will save or destroy a company. At that point, people mostly go to work to play politics. Occasionally, as a side effect of the political maneuvering, work sometimes gets done."
But at a 50 person company, or even a 500 person company, it's possible for single person to make a huge difference. And yes, if the company is competent, then saving the day will cut through the politics to a remarkable extent.
To some extent, this must also be true at a Fortune 500 company. But there, most of the people will the ability to "save the day" will usually be in upper management. The big tech companies may be an exception this: Google has their two "level 11s", and Microsoft had the massively talented team that they used to catch up to Netscape Navigator in the 90s.