I can understand the EL takes more matches to win, but it's also more valuable in terms of financial reward throughout, increased UEFA coefficient for future benefit, and ultimate reward of CL qualification. More matches but a much more beneficial competition.The idea that most supporters want to "bin off the EL" is a myth, the vast majority I've spoken to and read on here would love to win it. The main reason it gets moaned about is that there are 15 games between the group stage (where some will be against unglamorous minnows) and winning the final, including possibly matches against unfair CL drop outs. Whereas we can win the League Cup in 6 matches.
I'm not saying you can't want to win the LC, just that it doesn't make sense to support a cup run for the LC and not the EL. And honestly I'd be overjoyed with an LC win - wanted to win it last year. It's just the new sponsorship is embarrassing and makes it seem poxy...