Most Premier League clubs would indeed like to win the cup if they could be transported straight into the semi or quarter finals. But the earlier rounds and replays, often against lower league opposition who are going to give them a physical game are something they would rather do without. That's why very few of them field full strength sides.
Cup success or failure has no bearing on a managers job anymore, unlike in Fergie's early days when it stopped the club from sacking him. Winning it now means nothing if the team's league position isn't satisfactory.
West Ham beat Birmingham yesterday. I'd swap that for a point against Arsenal next week and another from Liverpool in early February.