For now, I decided to not auto-focus the search field. Mainly because it requires the user to swipe-down / dismiss the keyboard manually before being able to navigate to other pages (on Android it's easier, you can navigate back/away from the search page with the keyboard open).
Also, I checked other apps (from Apple: App Store, Music, TV) and they all don't auto-focus the search field when navigating to the search page.
However, on macOS, the search field will be autofocused in the next app update