As noticed by @jpnurmi here, GNOME's OSK does not come up when a text box is selected in a Flutter app. It works with Flutter on X11 and it works with normal GTK apps on both. May be related to zwp_text_input_v3. I'm looking into it.

EDIT: Flutter is sending some zwp_text_input_v3 related requests, but maybe not all the ones needed?? With GTK GNOME shows they keyboard in some contexts and not in others and I can't decipher it's logic. Maybe zwp_text_input_v3 isn't even the thing it's using to decide. More investigation is needed.