Not such a daft idea. Football is more an more driven by money, and the have and have nots are getting further apart. This is true in the Prem with most clubs (without major investment) really having no chance of challenging for the title. But also true throughout the various Divisions in the FA.
Cup runs bring in major cash, especially high profile compititions like UEFA run compititions. I'd like to see a revamping of the 3 major Euro cups, Champions League, UEFA Cup, Intertoto. Maybe making Intertoto or a new compitions, a cup lower division clubs can compete in would be in line with your idea. Being realistic is there any Prem team that would really want to win the Intertoto, but sure lower division teams, and their fans, would love to take a crack at it.