Kit looks solid. But I curious what shots you are hoping for that you can't get now. Is it reach that is the issue? Focus speed and tracking? Are you planting yourself in the end zone and trying to cover the game that way? What is the problem you are trying to solve? Need more ISO performance? You will eliminate hardware as an issue with what you are proposing. But you are also looking at spending a lot on your kit too.
For me, I hate teleconverters, regardless of price or make. It's just one more piece to worry about. Your 7d with a 300 will give your reasonable coverage. A used 1D mk IV with a used 400 2.8 will give you even better coverage due to its 1.3 crop sensor. I know its really old, but it was and probably still is one of my favorite cameras. It can easily shoot ISO 3200, and still performs well a stop or two even higher than that. That and a 1 Dx with the 70-200 should give you complete coverage of the field.
There are a lot of issues that even the best new camera can't solve at High School football stadiums. Light fall off in the end zones creates horrid shadows. I'm not aware of any camera that solves that problem yet. I don't like to see guys with strobes on the fields either - which can solve the problem. But just be really clear what the issue is your trying to solve. The answer my not be what you expect. More money helps... but not always.