### Godot version

4.0.stable

### System information

Windows 10, Forward+, GTX 1070 driver 528.49

### Issue description

When trying to edit the handle mode of a bezier cruve key from the inspector tab, the mode doesn't change. It works fine when right clicking on the key and setting the handle mode from here though. 

https://user-images.githubusercontent.com/24484962/222379540-29d71b6d-ff0c-4a21-9905-9750d0581fda.mp4



### Steps to reproduce

On an animation track using a bezier curve, select a key and try to change the handle mode from the inspector tab

### Minimal reproduction project

N/A
case Animation::TYPE_ANIMATION: {
			String animations;
			if (root_path && root_path->has_node(animation->track_get_path(track))) {
				AnimationPlayer *ap = Object::cast_to<AnimationPlayer>(root_path->get_node(animation->track_get_path(track)));
				if (ap) {
					List<StringName> anims;
					ap->get_animation_list(&anims);
					for (const StringName &E : anims) {
						if (!animations.is_empty()) {
							animations += ",";
						}
						animations += String(E);
					}
				}
			}
			if (!animations.is_empty()) {
				animations += ",";
			}
			animations += "[stop]";
			p_list->push_back(PropertyInfo(Variant::STRING_NAME, PNAME("animation"), PROPERTY_HINT_ENUM, animations));
		} break;
	}
	if (animation->track_get_type(track) == Animation::TYPE_VALUE) {
		p_list->push_back(PropertyInfo(Variant::FLOAT, PNAME("easing"), PROPERTY_HINT_EXP_EASING));
	}
}
void AnimationTrackKeyEdit::notify_change() {
	notify_property_list_changed();
}
Node *AnimationTrackKeyEdit::get_root_path() {