Rodrygo is still young and needs enough time on the field in order for him to improve himself which is actually his problem, because Real Madrid wouldn't give him that time. Moving to a new club will be the best for him but if he doesn't get an offer from the clubs of his choice, he have to stay in Madrid. Rodrygo should create more time working on himself.
I don't think Rodrygo needs to work on himself to impress Xabi Alonso or make Xabi Alonso change his mind and start using him at Real Madrid because since Rodrygo has fallen out of favor with Xabi Alonso, there is nothing Rodrygo could do at Real Madrid that will impress Xabi Alonso or make Xabi Alonso change his mind. Rodrygo is a good player, and he has helped Real Madrid in so many ways since joining Real Madrid, but if his service is no longer needed by Xabi Alonso, I think Rodrygo should move to another club where he will have more playing time.