Does the Courier actually take over and run New Vegas?...

While the dialogue leading to the end of the game seems to suggest that this is what will happen, the actual ending slides depict the Courier having little impact in the Mojave when compared to the NCR/Legion/House victory slides.

One example of this is if you weaken the Fiends by killing some, but not all, of the leaders then go for the House ending, he will send in the securitrons to wipe them out. However, in the independent ending the Courier does not take similar action and the Fiends remain as somewhat of a problem.

Actually Yes Man runs New Vegas. But he does so by following whatever orders the courier gave him. The Courier doesn't have total power in any ending, he just has massive influence.

But doesn't being able to give Yes Man orders mean that the Courier does have the power?...

That just shows that the Courier is lazy as hell. Most likely boozing it up in the casinos