Do I need an audio recorder?
Back in the day when movie cameras recorded to film, the sound made by the cameras was something to be isolated from the sound recording process.  Digital video cameras make very little noise (with the odd exception of things like early model RED cameras that needed high-powered fans to deliver sufficient cooling for their hot-rodded internals).

But that's less than half the problem.  The larger part of the problem is that to get good sound you need a proper microphone in a proper location, and that location is usually much closer to your talent than the camera will ever be.  As a rule of thumb, you want the audio microphone to be between 6-12" away from the mouth if it's a lav mic, 12"-24" away if it's a typical cardiod vocal mic, and 18"-36" away if it's a shotgun (boom) mic.  And in all cases you want the mic oriented toward the speaker (protecting it from plosives, of course).

It is for this reason that you want to divorce audio and video recording and then remarry them in post-production.  In that case, the audio recorded on the video camera will give you a very good idea about how to perfectly line up the audio from your audio recorder, and once you have done that you can mute the audio from the video camera and use the audio from the audio recorder.

If you shoot 10' away from the subject and try to record audio from the camera's location, it's going to sound terrible, no matter how much you spend on microphones (unless you also spend $1M+ on an amazing acoustic environment).

Here's a link to a video that really goes over this in detail.

And of course if you do have fan noise you want to avoid, or other sources of noise, that's also a reason to move the microphone, and the talent, away from those noise sources.  If the noise sources are intrinsic to the room (such as a refrigerator you cannot turn off), then you begin to understand the logic of why people rent sound stages...