The League Cup is like a relative you do not get on with, but still have to ask to Christmas lunch. You would rather they were not there, but there is nothing you can do about it. And when they arrive you do little to make them feel wanted.
Premier League clubs would be far happier if the Capital One Cup did not exist. It is no more than an intrusion into the real thing, "league football," and they make it blindingly obvious that they take part only because the rules state they must. Going out of the competition is almost reason for celebration; advancing is a burden. Another unwanted fixture awaits.
Most Premier League clubs can realistically win only two trophies each season, the Capital One Cup and the F.A. Cup. Only three or four heavyweights contest the title or have their sights on a run in the Champions League. For the rest, it's the domestic cups or nothing.