Steven, some generalized thoughts that might be completely useless...
1. That lens/tc combo might be impossible to stabilize short of a steadicam or a broadcast-style stand. The angle is just too narrow.
2. Video AF for any camera on the market is simply not there yet and has little to do with technique. It's useful for stationary vlogging but that's about it. All other circumstances, It'll hunt and ruin a take. The biggest, most expensive rigs are MF. Lots of YouTubes on how to MF.
3. Not sure how much cropping you did in post and what the settings were, but if possible, ridding the TC, shooting 8k and cropping to an HD might work better. Even shorter lens.
4. Finally, in a situation like this, I would sometime shoot a long take, as you did, but then chop it up in post into really short segments with quick transitions. This will allow you to rid of OOF frames or overly-Jerry footage, but still giving you a continuous sequence. Just mind that the birds are roughly in same place or moving in the right direction.