Unfortunately flashing Android 12 to your Xperia 10 III will not fix echo cancellation issue. Discussed also over here. Further, I do not believe that disappearing sound and freezing video would get fixed either.
From my past experience it has been actually other way round. When developers have (back)ported something for a device adaptation let’s say for wlan driver, then when upgrading beyond the recommended Android version the wlan driver started acting up as the adaptation fix no longer worker.