I shoot a 300mm f/4 (PF) and 1.4x TC all the time and I'd agree 1/2000" should be plenty fast enough to limit camera shake issues though not always fast enough to stop motion blur issues for things like flying birds.
But the thing is, everyone's handholding ability is different and can depend on many things. You've mentioned you have a lot of hand holding experience, have you had better luck handholding your other lenses at similar shutter speeds? IOW, are you getting sharp images at 1/2000" with your 200-500mm racked out to 400mm or 500mm but not with your 300mm AF-S lens plus TC? If so that might come down to the VR in the newer lenses including the 200-500mm giving you a bit of an edge and compensating for some camera movement.
Also are you sure it's camera movement and not soft images just from the 300mm + TC combo? Do you see characteristic motion blur/shake issues like catchlight in a subject's eye becoming elongated or subtle ghosting around sharp edges or is it just an overall softness to the images? If the latter it could be either AF errors with the lens plus TC combo or the older lens plus TC combo may just be a bit soft compared to your 200-500mm without TC. Have you checked for front or back focus issues with the 300mm AF-S lens plus TC?
I guess the common sense thing to try is just shooting the 300mm + TC combo at faster shutter speeds in good light to see if your image sharpness improves. If so, then you've more or less answered your own question and might need to seek support more often (even something as simple as taking a knee or leaning up against a tree or fence can add a lot of shooting stability) or work on the small stuff like rolling and not jabbing a finger at the shutter release, keeping the elbows tucked in tight and always seeking a very stable stance if you're not already doing all those things.
But personally I shoot a lot of 300mm + 1.4x TC images at much slower shutter speeds than 1/2000" for static or slow moving subjects though these days I am doing that with VR equipped lenses. If shooting at even faster shutter speeds doesn't sharpen up your images of static or slow moving subjects then it's quite possible it's not really motion blur you're seeing but perhaps overall image softness that might be improved with AF Fine Tuning.
Also, just to put it out there if your sharpness and blur issues are with fast moving subjects like flying birds then I'd definitely bump up your shutter speeds a bit. FWIW, I start at 1/2500" for flying birds unless I'm intentionally trying panning at slow shutter speeds for background blur and often shoot faster than that for quick flying birds with 1/3200" and 1/4000" not being unusual.