I use AF 'zone' rather than AF(L), it seems to be more reliable when dealing with birds in flight that are reasonably close.
There's movement in multiple directions with a big bird flapping, wings up and down whilst the whole thing is flying forwards, makes for hard work for the camera.
The extra shake certainly won't help!
I'd try AF-S and spot a/f. If the percentage of correct exposures goes up (albeit the overall number of shots may reduce) then it's likely to be a technique problem. If the focussing issues continue then maybe your lens has become a bit stiff
The K-3iii can (almost) be it's own worst enemy. With so many configurable features it does take time to find the best combination for any one person and any one set of circumstances.
Fortunately, judicious use of the SFn wheel and maybe a few custom User Modes can allow multiple configurations to be tried "on the fly".
I have my SFn wheel set for AF-S 'spot', AF-C 'spot' and AF-C 'zone' and regularly flip between them with the camera to my eye.
I'm even getting used to adjusting the over/under exposure with the +/- button using just the viewfinder.
The K-3iii really is an ergonomic delight, but there is a learning curve
Enjoy