No, it does not. Not according to the physics as I understand it.
A sensor's noise is best considered in terms of the amount of noise per square millimeter of sensor. No matter how you slice up each millimeter, the noise is essentially the same. In other words, doubling or halving the number of pixels per millimeter will not change the noise in each millimeter of sensor.
Regardless if you use the TC or not, you end up with the same noise, especially in a very high ISO situation like you have here.
When you take off the TC, you get to increase exposure by one stop, and that is the obvious benefit to doing so.
What is somewhat hidden under the covers is that by shrinking the image by 1 stop of focal length, you are increasing the noise when you have to stretch those square millimeters of pixels more in the final output.
You have a D800. If you shoot a properly exposed ISO 6400 image and then display or print it at some reasonable output size I suspect it probably looks pretty good. Even if it has some challenging tones in it, like those OOF blurs that can look so ugly in a lesser (older) camera.
Now if you look at that image at 100% pixels I suspect it looks pretty ugly. At least that is the comment I have heard a number of times from "pixel peepers" who do the same.
The difference is the magnification.
You are doing the same thing when you drop the TC and go with DX. The image size on each sensor, with the same lens and focal length, is the same size and it contains the exact same amount of noise. When you drop the TC and go with higher density DX, you get the same pixels across your cropped final output but you then add in a stop of noise because you are then increasing the magnification of the sensor (on a unit area basis) by 1 stop!
The result is a net push, or it should be. If it is not, then all the physics behind sensors, and the basic theory of "unpredictable photons" (shot noise) needs to be rethought.
Here is another way to look at it. Your 300/2.8 has a front element of 107mm diameter. If you calculate the area of that element that is called the "light gathering power" of the lens.
No matter what you do with that lens, no matter how you stretch the effective image across the sensor with TC's, and no matter how many pixels you pack into that sensor, the same number of photons hit the sensor across your final cropped subject frame.
The theory of imaging noise says that while a certain number of photons *should* hit each sensor well (pixel), the actual number will vary, and that is shot noise. It is a statistical game where the more photons you drop onto that pixel, the more consistent the actual measured result (less noise).
The key here is that the same is true if you look at the amount of light that hits the sensor per square millimeter, or any unit of measure you prefer.
In the final analysis, if all else is equal between the DX and FX sensor (they have the same noise per square millimeter and they will if they were designed around the same time) there is one and only one way to reduce noise in the case where you have a fixed size subject and a fixed working distance: make the front element of the lens bigger.
And it doesn't matter what focal length lens you start with, or end with after adding or subtracting TC's. The amount of light across your intended final crop that hits that sensor is the only thing that matters, and that is controlled purely by the size of the front element.
When it comes to noise, there is simply no free lunch. Lenses more or less cost so much per square inch of front element, and they weigh so much per square inch of front element. And those square inches of front element control the noise level.
All the above is why I previously said that I am "old school" and that I do not believe you can scale up sensors indefinitely and transparently trade focal length for pixel density. And why I have trouble with your idea that the Pros are all going to small lenses and DX cameras.
It defies physics. It only works if the cameras have "excess high ISO performance" to the tune of a full stop.
You have, in different replies here, suggested FX does and doesn't have excess high ISO performance, depending on the context and shooting situation, which makes it hard for me to keep up with a moving target . But once you show me a 3 stop underexposed image as your best way to cope with a real world situation, I gotcha!
You can certainly change the FOV via high density DX to deceive yourself into thinking you have a "longer focal length" and if your long lens technique is beyond reproach you can (in theory, avoiding diffraction issues as you generally can when shooting wide open) transparently trade pixel density for focal length and get equivalent resolution. There is sort of a free lunch here, but only in regard to resolution in terms of pixels across the subject.
But there is no free lunch when it comes to noise. Noise is solely a function of your front element size, and nothing else .
The above assumes all else equal, that the sensors are identical in terms of performance per square millimeter.
But consider that in a reply elsewhere here today you suggested that FX is near ideal but DX still has some room to improve. If that is the case, then you are also suggesting that the DX/FX trade is even worse than the push I suggest above because the DX sensor inherently does not perform as well as FX on a per square millimeter basis - which is how the "ideal performance" is computed.