Gear upgrade from T2i - RF, EF, non of the above, all of the above

'Ello everyone! I (16yo, jobless, birb lover) am planning to upgrade my gear to some nicer stuff finally. At the current moment I am working with an EOS Rebel T2i and EF-S 55-250mm lens that I got second-hand from a relative. This paring has served me very well in the few years that I've had it and I've managed to get some really nice photos with the setup. I primarily take bird photographs, so distance and cropping are two things that I encounter quite frequently. For that reason I'm thinking that a setup with some more range and higher megapixel count would be ideal. Budget is of course an issue as it is with everything, and I'm going for something in the range of about $2k, which I should also add probably isn't stretchable at all because this is already with an addition of parental funding and I doubt they really want to spit out more than like $600 right now, understandably. The gear I'm looking at currently would be a used R7 from ebay + an RF 100-400mm, or the same body and an EF 100-400 ii with an adapter. I have never used auto focus on the setup I have now because it is absolute trash and can't find a bird in a million years, so all the good things I've been seeing about this new combo is really appealing. My question I guess is what assortment of this stuff would be most beneficial for the amount of money I put into it. Should I get the R7 and RF 100-400, or should I try to convince my creators that getting the L series EF lens would be a huge improvement (and would that be truthful)? Is getting the R7 unnecessary if I get the EF, because I can get the same quality with it on my T2i because the lens is more important than the camera? Or is the somewhat famous budget-build R7 and RF perfectly amazing and the higher aperture that comes with it won't be a problem? Is getting a new lens a waste of money and the 55-250 I have now completely sufficient if I get an adapter (although the extra reach of one of the others would be nice)?

And then finally, just overall is any of this stuff going to be a substantial improvement over what I have right now? I know that good photographs come from the person behind the camera, not the camera itself, but I am not a professional photographer, and I do not have the skills of one, so having a camera that is more easily capable of higher quality shots would be very helpful.

Thanks to anyone who takes the time to read my Great Wall of Text! Hopefully I articulated everything well and didn't ask anything too dumb.