I think Rodrygo will stay at Real Madrid even though he will likely only be a substitute in the future, that's because Man City still hasn't found a clear point regarding Savinho agreement with Tottenham and as far as I know Man City is currently more focused on recruiting Donnarumma, with the transfer window closing soon, it means they don't have enough time to recruit both of them.
I do not support Rodrygo to remain at Real Madrid this season because it is very clear that he will not be having enough playing time in the team as the coach is not including him in his plans for the team. If Manchester City is focused on bringing Donnarumma to the team the best thing if for him to look for another team that will sign him while Real Madrid will reduce the amount placed on him if they want him to be sold out.
Sincerely I agree with you, Rodrygo is too talented to be wasted on the bench for Real Madrid, if it's obvious that Xabi Alonso doesn't want him, then he shouldn't keep him to bench, at best he should be sent on loan, since the amount placed on him by Real Madrid is high and a bit scary to some clubs. I strongly doubt if Manchester city had an open bid for Rodrygo, even if they did, how strong was it, i think it was just mere speculation. The caveat to which Real Madrid is looking at is the fact that the youngster is still at his prime so they wouldn't want to sell him at loss, so they want to cash out too.