That might have been true a decade or so ago but the Premier League teams are taking the League Cup more and more seriously. Look at the last ten finals:
2012: Cardiff City 2-2 Liverpool (2-3 Pens)
2011: Arsenal 1-2 Birmingham City
2010: Aston Villa 1-2 Manchester United
2009: Manchester United 0-0 Tottenham Hotspur (4-1 Pens)
2008: Chelsea 1-2 Tottenham Hotspur (AET)
2007: Arsenal 1-2 Chelsea
2006: Manchester United 4-0 Wigan Athletic
2005: Chelsea 3-2 Liverpool (AET)
2004: Bolton Wanderers 1-2 Middlesbrough
2003: Liverpool 2-0 Manchester United
19 of the last 20 finalists have been from the Premier League, and maybe 8 have been what you would call 'major'. The bigger clubs are taking it more seriously. It's not a particularly exhausting competition. One game a month and with it all done by March does not constitute a reason why England are rubbish.
Just because the finalists/winners of the carling cup generally are the premiership teams means that they take it seriously.
I would think even if those big league clubs put out their non first xi to play the championship, league 1 & 2 sides during a carling cup match, they would still generally be able to beat them - and that is what generally happens.
Lets just use Man City this season as an example.
They played Arsenal in the league on sept 23 and drew 1-1, just after losing mid-week, 3-2 to madrid in the c-league on sep 18. Then on sep 25th they had to play Villa in the league cup and lost 4-2 @ et.
I don't think its unfair to say that City didn't bring their A-game to that match and were glad to lose it and get knocked out and not have to bother about it again this season, as they seek to focus and keep their players fresh for the title defence and a better champ-league performance in the coming months.
Fixture congestion caused by the useless carling/captial one cup in the English league is one of the reasons most certainly why England are poor. It is proven fact, off all the major European nations that goes into world cups and euro's since world cup 2002 - England have the most injury woes that any other team.
In EURO 2012, England lost two mid-fields to injury in Lampard, Barry, Wilshire, Cleverley and played with a crocked Parker next to Gerrard. Spain, Italy, Germany, Netherlands, Portugual had no such major injury woes at once.
But its not the main reason quite clearly, so lets not loose track of the main point of why simple the winter break that everyone has is needed in this country . Lack of football intelligence in heath of battle, inability to pass the ball properly and poor production line of players are the bigger issue.
All of these area's have to be improved along with the winter break implementation, for England to play better on the world stage.