I look at it this way - say you have a business and you have three clients. One pays you £1,000,000 a year if you hit attainable targets (The League), the other pays £300,000 with an extra £200,000 if you hit difficult targets (The FA Cup) and the other pays £50,000, with an outside chance of a £200,000 bonus if you hit every target (The League Cup).
Which one are you going to pay the most attention to and put your valuable assets towards achieving?
Everton is a business, whether people like it or not that's a fact. The focus will always be on the primary competitions. You're 100% correct in what you're saying, it's common sense and actually goes beyond opinion.
The difficulty is that the actual issue of a "trophy" is an emotive subject, so business logic and common sense goes out of the window at the expense of a lust for a trophy - hence artetafan saying he's rather lose half a dozen league games than one League Cup game. Logically, that is an incredibly bad opinion, but if the allure of silverware is that strong then as a football fan that is what is going to take priority.
But... It's the League Cup, so I still can't understand it. If it were the FA Cup, then I'd completely understand emotion overpowering logic, but for the League Cup the whole thing feels like overkill for me. We played a shadow side like every other club did and we got put out by a Premier League club, I don't understand the drama.