You don't necessarily need a $2000+ camera. As you said, the photographer is the most important part, however good cameras make life a whole lot easier and allow shooting in situations you otherwise couldn't. Particularly for low light, having full frame makes a big difference. Additionally, higher end cameras give more direct access to controls to make adjustments on the fly.

The D5200 is an extremely limited entry level model. It is not designed for rigorous use or rapid adjustment. It is designed for casual shooters that want to step up to having better optics. It has a limited user interface and very limited capabilities compared to even a midrange camera.

If you are good enough and can use a flash, then you can probably get by with a more limited camera and being able to anticipate your situation enough in advance to adjust the camera as needed, even if it is harder to control, but having that extra light sensitivity and controls right at your finger tips can save photos you would otherwise miss. Similar can be said for lenses. You can use a cheap prime lens to help with low light, but you are limiting your ability to catch shots quickly by shooting a wedding with a prime. It is doable, but far riskier and trickier to do well, with far more chances of missing key shots.

For weddings in particular, I would say that shooting with a mid-range model is certainly possible, but I would personally advise against going with a low end model for weddings unless you are a second shooter. I would also suggest that the high end model cameras and fast zoom lenses (even if third party) are well worth the cost and will make your life much easier for weddings.

Outside of time-critical and low light uses there is far less need for a higher end professional camera. Again, they are still useful to have, but you don't by any means have to have one to do professional level work. The biggest single quality reason for buying a higher end cameras is the shallower depth of field and additional detail you get from full frame. The biggest single advantage to usability is the added control interfaces.

(To give you an idea of my background, I first entered the DSLR space with a Canon xTi with a 17-40 f/4L lens as my main lens. I eventually upgraded to a 5D Mark iii and f/2.8 zoom lenses in order to start shooting weddings. I would consider using something like a 70d or 7d as a backup body and it could probably be used as a main body, but I'd never use something in the Rebel (#00D) line (equivalent to Nikon's 4 digit model numbers) for such situations. Certainly not as the primary shooter. It is just too limiting and too risky. You don't want to screw up capturing the most important day of people's lives to that point.)