I don't think you can just call it an admission of failure. Yes in the first 10 years of his arsenal career he treated it like a second rate competition but apart from the odd year look at the teams who were winning it..
Villa, Leicester (X2), Spurs, Blackburn, Middlesborough, Chelsea (when they were very good but not great).
Even before then we have Liverpool (when they were not a top team), Villa, Forest (x2), Luton, us (when we hadn't won anything for ages), Oxford and Norwich.
From 1984 to 2005 the League Cup was a team for the second and third tier teams to fight it out for a win. It is not a coincidence that it was those level of teams that were winning it in that time period, they were winning it because that is the kind of competition it was.
Now look at who has won it since 2005
Chelsea, Man Utd, Chelsea, Spurs, Man Utd, Man Utd, (and hopefully now us).
The competition has changed, just like 1977 to 1984 it is a trophy for those who are challenging for the title and in the Champions League. Wenger is responding to that and taking it more seriously.
The fact that we haven't won anything for 5 years comes into it of course, it would be naive to think otherwise, especially with the pressure Wenger is under from the fans but I don't think that is all there is to it. The competition has changed in this half of his managerial career.