### Godot version

4.0.3-stable

### System information

Windows 10

### Issue description

In some projects ViewportTexture remains pink and doesn't render viewport content after upgrading from 4.0.2-stable to 4.0.3-stable. If a pair of files viewport.h/viewport.cpp are downgraded back to what they were in 4.0.2-stable, problem goes away.

This is how it looks in 4.0.3-stable:
![4 0 3_stable_viewport_4 0 3](https://github.com/godotengine/godot/assets/1449921/44c191a9-4b67-4b02-90b6-4ab615f62a73)

And this is how it looks if viewport.h/viewport.cpp are downgraded back to what they were in 4.0.2-stable.
![4 0 3_stable_viewport_4 0 2](https://github.com/godotengine/godot/assets/1449921/f5abc605-f129-4222-9c0e-e9a07621dd9d)



### Steps to reproduce

Unfortunately, I cannot come up wit a simple demo project reproducing the problem. It exists in some projects and doesn't exist in others.


### Minimal reproduction project

Cannot produce a simple project reproducing the problem. In simple projects it works fine.
ERR_MAIN_THREAD_GUARD;
	WARN_DEPRECATED_MSG(R"*(The "push_unhandled_input()" method is deprecated, use "push_input()" instead.)*");
	ERR_FAIL_COND(!is_inside_tree());
	ERR_FAIL_COND(p_event.is_null());
	local_input_handled = false;
	if (disable_input || !_can_consume_input_events()) {
		return;
	}
	if (Engine::get_singleton()->is_editor_hint() && get_tree()->get_edited_scene_root() && get_tree()->get_edited_scene_root()->is_ancestor_of(this)) {
		return;
	}
	Ref<InputEvent> ev;
	if (!p_local_coords) {
		ev = _make_input_local(p_event);
	} else {
		ev = p_event;
	}
	_push_unhandled_input_internal(ev);
}
#endif 
void Viewport::_push_unhandled_input_internal(const Ref<InputEvent> &p_event) {