Manchester City and Real Madrid always give a lot of importance to their domestic league and Champions League but we don't see that in case of Arsenal. Arsenal may take the English Premier League tournament very seriously but they are not too interested in the Champions League tournament. If they had given more importance to the Champions League tournament, their performances in the Champions League would not have been so bad on a regular basis, while in the English Premier League they are rightly second in the points table. Real Madrid and Manchester City are as much expected in the Champions League as we are in Arsenal, so Arsenal should prepare for the Champions League and prepare accordingly.
Real Madrid will certainly be ahead of the rest of the teams next season with the way the teams have been so far and Manchester City could do well there as well but Arsenal have yet to change their squad so it wouldn't be a surprise if their performance is similar to last season.
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.
In fact, the priority for every big team is to win the Champions League. Because they have definitely won the league they play in the past. Reaching the championship in a tournament with the biggest teams in Europe is the dream of every team and every player. This is Real Madrid's priority and this is Manchester City's priority. There were many teams trying to win the Champions League in every aspect, especially after they no longer had any claim in the league. The latest example of this is Borussia Dortmund last year. They left the league and made it to the Champions League final, and they almost won.