I use a Sony AX700 and recently experienced a huge improvement in autofocus success underwater. A diver heard me complaining about focus failure using WIDE, and suggested I try CENTER focus. Oh, what a complete difference it has made!
Here's a bit of detail:
Including CENTER, there are five focus modes available for the AX700 camera in video mode. The other four are MANUAL, ZONE, FLEXIBLE SPOT and WIDE.
ZONE and FLEXIBLE SPOT are available if the screen can be touched or if the joystick can be operated to indicate to the camera where to focus underwater. These aren't available on the Gates housing.
So until recently I used WIDE, following the recommendation of the "experts." This resulted, as you mentioned, in lots of focus hunting and failures to focus. Very frustrating.
When I switched to CENTER focus, magic seemed to happen! Most of the out of focus moments went away - my footage is in focus probably well over 90% of the time. What a great relief!
Hope this helps,