The only big incentive to get clubs more interested in winning it would be to offer a Champions League place for the winners, and likewise for the FA Cup. Problem is, then you'd have to take away the 3rd and 4th place CL spots and only give to 1st and 2nd, and the top 6 or 7 clubs would never support that.
The league cup is one of those funny tournaments where if you get through, fine, if not, who cares? It only starts to get serious at the quarter-final stage. I'm sorry, but cups are always going to be of less importance than league games because winning a cup won't secure your long term, or even short term, future. Look at Birmingham, they've won the cup, but they also got relegated. Now they're in financial difficulties and facing a few years at least outside the Prem. There's nothing that will help to revamp the competitions, other than offering league points for wins or some kind of relegation immunity, and that won't work so the cups have to take a back seat.
I would support it. Why should you have a chance of being European Champions when you only finish 3rd or 4th in your domestic league? It would never happen but it would help the competition no end.
