Exhibit 1.
This morning I took Murphy for his morning walk when a small flock of Lesser Goldfinches landed in a tree about 30 feet away. Without moving I dropped the leash, brought up the trusty D7200 with the 70-300 VR lens and fired off 23 shots in short bursts. Feeling confident that one of the shots would be a keeper, or could be turned into a keeper in PP we went on our way. None of them could.

Exhibit 2.
Earlier this year Murphy and I found a bright red male Cardinal in an Ocotillo plant. I fired off about 50 shots in short bursts from where I stood. About 10 of them could be called keepers except the Ocotillo stem was growing right out of the top of the Cardinal's red crest. No problem, I took them out in PP.

There are many more instances of careless and laziness I could catalog. I have become a "Spray and Pray" photographer all because in digital photography "pictures are free"! Back in the film days, I don't remember exactly because I'm old and forgetful, but buying and processing a 36 roll of Kodachrome was between $10 and $15, and you had to wait a week to ten days to get your pictures back from Kodak. I was careful in my stalk, bracketed my shots and each set of 3 cost me about $1.

Conclusions
I am not as good a photographer as I once was! Free pictures and improving PP skills have encouraged me to become sloppy. The only thing I can't fix is focus and I would not be surprised if that coming soon.

While the P900 is a nice camera I would say not a great camera for sports. Especially indoor sports. I can’t speak to its video capabilities as I don’t shoot video. I don’t think the P900 would be an upgrade over your D3200. As far as I know it does not shoot raw which you really need for inside gyms or rinks. The other limiting factor is the burst rate. While is can shoot up to 7 frames per second the buffer can only handle 7 frames then it takes a couple of seconds to clear the buffer. That is really limiting.
Burst rate and more important buffer capacity does matter for action sports. I believe your D3200 also has better auto focus than the P900 which is another important factor shooting sports Unfortunately sports is the one genre where equipment does matter. You need fast lenses and a body with a decent burst rate and buffer and good auto focus. I have shot sports (hockey and lacrosse)with a D3100, D7100, D7200 and D500. The D500 (3x your budget)is made for sports and my personal opinion is the D7200 ($150 more than your budget) is the bare minimum you need to shoot sports. while it is a great camera it is still very limiting for sports.
To give you perspective I am not a gear snob. I am a budget shooter I love finding lower cost solutions to photography. I have successfully shot ice hockey and had images published with just a 50mm and 85mm 1.8 on the D7200 I do not own a 70-200 2.8 although I have on occasion rented one. I have had lacrosse images published using a $150 70-300 AFP lens with the D7200. So while I don’t think you need to spend crazy money to shoot sports I think you may need something better than a P900. As for video your D3200 has better video capability. Over all I am not sure where the P900 would be an upgrade. In my opinion the best upgrade to the body you have right now would be lenses
I hope this perspective helps rather than muddies the water for you
Good luck.