Is this specific to the A7SIII? CIZ in video mode doesn't disable AF on my cameras (A9, A6500), but face af isn't available, so I guess it's contrast based. It works well though, I use it all the time.
Controlling CIZ remotely works on the older cameras with apps, but not on my A9. Although it might on other newer models. This chart (referring to compatibility with Sony's desktop software) shows how the original A9 lacks certain features, https://support.d-imaging.sony.co.jp/app/imagingedge/en/devices/