Films where I don’t recognize a single actor among the whole crew are almost always better than ones where I’ve seen such and such actor in other movies. Just more immersive. And even if they’re not the best actors I’d much prefer that over whatever the hell Chris Prat or Tom Cruise or Leo D are up to.
If your camera supports it you can try registering faces to prioritize the auto focus for a few of the standout players.
If your camera doesn’t have any ibis and you’re shooting with a crop at 200mm I think maybe you’re right that 1600 isn’t quite fast enough. You could go higher or maybe try out a monopod to help stabilize.
There should be a tracking AF option where you can select a focus point and then let the camera track the point but your camera may not have that option either.
If no tracking AF try using a larger point. My camera calls it zone AF where it focuses on one of the 9 boxes made by the lines of thirds.
It could also be a “get good” situation where you just have to anticipate a little of what might happen and try to be ready for shots a few seconds ahead of time. Practice makes perfect after all.
I hope that maybe this helps!