They need to look at the own settings in xbox settings → privacy and safety. Could be this setting: “Others can communicate with voice, text or invites”.
Try logging out completely of your account and restarting minecraft. Then log back in. Then restart minecraft again. If it’s in your microsoft purchase history - you have it.
Is it the side to side motion of the cape you are calling a jitter? In the videos posted are you only moving directly forwards with no left or right?