I've got one mate who supports Liverpool and he even said the day before the final day, even when he knew they weren't likely to get it, that he'd rather have the Premier League. I think most of them would, because the Premier League is the better measure of which team is the best. Cup competitions by their nature have a large element of luck involved. When you hear people talk about the great teams of the past, it's always based on their league performance, rather than the Champions League (unless they win both in the same season, or they do three back-to-back like Real Madrid). Nobody points to Liverpool's 2005 team or Chelsea's 2012 team as one of the greats of English football. Mourinho's team that dominated the Premier League are considered among the best though.
You want to win the CL to show you're the best team in Europe (and therefore the world).
You can't be the best team in the world if you finished 2nd in the league.
That's why they care so much. Being crowned Champions of Europe when you know you aren't is hollow.