### 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.
if (Object::cast_to<InputEventKey>(*p_event) != nullptr || Object::cast_to<InputEventShortcut>(*p_event) != nullptr || Object::cast_to<InputEventJoypadButton>(*p_event) != nullptr) {
		ERR_FAIL_COND(!is_inside_tree());
		get_tree()->_call_input_pause(shortcut_input_group, SceneTree::CALL_INPUT_TYPE_SHORTCUT_INPUT, p_event, this);
	}
	
	if (!is_input_handled()) {
		ERR_FAIL_COND(!is_inside_tree());
		get_tree()->_call_input_pause(unhandled_input_group, SceneTree::CALL_INPUT_TYPE_UNHANDLED_INPUT, p_event, this);
	}
	
	if (!is_input_handled() && (Object::cast_to<InputEventKey>(*p_event) != nullptr)) {
		ERR_FAIL_COND(!is_inside_tree());
		get_tree()->_call_input_pause(unhandled_key_input_group, SceneTree::CALL_INPUT_TYPE_UNHANDLED_KEY_INPUT, p_event, this);
	}
	if (physics_object_picking && !is_input_handled()) {
		if (Input::get_singleton()->get_mouse_mode() != Input::MOUSE_MODE_CAPTURED &&
				(Object::cast_to<InputEventMouse>(*p_event) ||
						Object::cast_to<InputEventScreenDrag>(*p_event) ||
						Object::cast_to<InputEventScreenTouch>(*p_event)
								)) {
			physics_picking_events.push_back(p_event);
			set_input_as_handled();
		}
	}
}
void Viewport::set_physics_object_picking(bool p_enable) {