Your question is an interesting one to me because I'm having the same issue. I've been thinking maybe it was me rather than the lens/tc combination. I also tried AFFT but didn't have much luck. It is very disappointing since the tc was $500, and was advertised as compatible with the 200-500. Please let us know what you end up doing - I may follow your lead. Good luck!
If you own the TC and want to make it work with the 200-500, only use it with very good light. Use the center AF point - or if you are using the D850 or D5/6 you can use the adjoining AF points. Be sure to use long lens technique with a tripod and a hand over the lens barrel to dampen vibration. Don't even consider a cable release without a hand on the lens barrel. Use this combination for reasonable distances - inside 100 yards - not for distant subjects. Forget birds in flight. You might have some success at less than 500mm.
The TC works very well with the 200-500 and the Z cameras. You can eliminate shutter movement with Silent mode, and you can focus anywhere in the frame as long as you have adequate contrast. AF fine tuning is not needed.
It's not the lens and teleconverter that is the problem - it's using it with a DSLR creates some issues and the high magnification requires great technique.