I don't think that City are so fed up with EPL titles that they are ready to put the Champions League higher in their priorities. Maybe if they had a guaranteed choice between these two titles, they would choose the Champions League, but there is no such choice, and even reaching the final, you can lose everything. In terms of reliability, the EPL looks much better, especially when your team is in the lead, albeit by a small margin.
City and Liverpool both have busy schedules, they have to play in 3 different competitions, UCL, EPL and FA Cup. Perhaps, the trophy that is not a priority is the FA Cup. the rest they will try to be able to bring the remaining two trophies to the clubs. I'm not sure Liverpool only want to focus on one trophy, neither does City. They will focus on 1 title if they have been eliminated from the FA and UCL. if not, it looks like UCL and EPL are targets for them.
I think the first priority is the EPL. As for any major team from the top 4 championships. Titles in championships starting from 5th are evaluated differently and I am sure that any champion of Portugal or France would give 2-3 titles for the Champions League.
The second priority is the Champions League. This cup is valued less than the national championship, but only because this result is somewhat random.
The third priority is all other national cups and stuff, such as the club world championship. This is nothing more than a nice bonus to the main achievements, but if the main titles are lost, then such titles do not change anything.
I agree with you. I think all coaches will prioritize the Premier League and Champions League.
I think Manchester City is like that too.
I believe Liverpool will fight to the death to secure its position in the English league. In order to get full points against Manchester city in the English league, substitute players will be fielded when dealing with Manchester city as well as in the FA Cup match.