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.
The rivalry between Man City and Liverpool takes place in all leagues, as you said, not only in the Champions League which is a prestigious event between the two clubs, but the EPL and FA Cup are also still an opportunity to show that they have become a giant club. For Manchester City, while they have many gaps to take a perfect share in winning the trophy, inevitably it will be very possible to fight for it, as well as for Liverpool whose ambition until now is difficult to prevent.