There are a few things to consider -
First, 'fast autofocus' isn't really the crucial factor in photographing birds in flight, in general. Acquiring focus on a moving subject quickly is important, but more important is the continuous focus ability - can the focus system continually adjust the focus as the subject flies along and you pan with it.
Which brings up another consideration - there are two kinds of movement relative to the focus system - moving across the frame with focal distance not changing much or at all, or moving directly towards or away from the camera, where the subject's distance is changing constantly. Most cameras can cope with situation 1 - as once the focus is acquired, the subject isn't really changing distance relative to you. Situation 2 is much more complex, and this is where a lot of cameras will fall down.
With a bird in flight, some cameras add a 'tracking' function - this is where the focus system attempts to lock onto a subject based on shape/contrast/color, and stay with it anywhere in the frame - as long as you can pan and keep the bird inside the frame, the focus can stay with the subject and apply the continuous focus to it - this frees you from having to keep a specific focus point or area right on the subject or in the dead center of the frame.
The RX series, starting from Mark III, added Sony's very good on-sensor PDAF focusing - while probably not quite as robust as some of their interchangeable lens, larger sensor cameras, it still seems to do a solid job keeping up with BIFs. It will acquire quickly, continuous focus adjusts well once the subject is on the focus points, and the tracking system is OK at staying with the bird. You should have no difficulty with birds flying side to side...and will have some success, though maybe a few misses, with subjects approaching quickly at the camera. Most reviews consider it the most capable fixed-lens camera in this regard, though others can function acceptably with BIF work, such as the Panasonic FZ series with 1" sensor. The Sony would probably edge all others out with approaching/closing subjects.
I've shot with the RX10IV, but do not own it. I found it acceptable and capable for BIF work, though I am not sure it would be quite as good as my interchangeable lens Sonys with real-time AF tracking, and with small, fast, erratic birds closing distance rapidly.