I think you're honestly wrong, in my opinion Real Madrid gives more importance to the Champions League, in fact the season in which Barcelona won (two years ago) they actually abandoned La Liga to focus on the Champions League. I don't know about Manchester City, but for Real Madrid the priority is the Champions League.
Real Madrid will always pay attention to any competition, but it is true that they pay more attention to the Champions League because, when you think about it, the Champions League is one of the biggest leagues that people pay a lot of attention to. Winning the Champions League means you are not an ordinary club because if clubs from different leagues compete and you win out of all those clubs, it means you have level. Before this season they allowed Barcelona to win the la Liga and even when they focused on the champions league they still did not win but last season they were able to win the two at the same time this shows how good Real Madrid can be, Manchester city also put a lot of seriousness into winning, instead of Real Madrid fighting with Barcelona Manchester City have now occupied that position.