Upload & Sell: Off
I shoot baseball a lot and use a t2i and 7D, and when I do fast speeds at night games, the noise is really discouraging. I feel you on this.
Then I got a 6D and although I miss the FPS of the 7D, the image quality using fast shutter speeds in low light is seriously on another level. I barely get any real world difference with the differing AF systems and nailing focus. The 6D can do the job well in that regard despite it's AF system being "inferior".
If I were you, I'd get a 5D3. Unless you absolutely need to do sequence shooting that requires 10-12 FPS, the 1D bodies won't gain you anything.
If you have the money and it doesn't matter, I'd go for a 1DX. There's no reason to not have the best if you can afford it. I don't shoot for a living so I would never be able to justify buying that camera.
But, if you are also considering lens options, the 5D3 should deliver to your expectations and needs. Also, IQ is going to be head and shoulders above what you are used to which you will fall in love with.
My vote given your situation is a 5D3.