I am inclined to agree with you, but that puts things at an impasse. "Big" teams are already playing too many matches and the FA Cup is not held in the same regard as European competitions (and these continue to expand) as well as international tournaments. Even so, the big teams will keep winning the cup, because even when they treat it as a secondary priority they have the talent to win it anyway 9 times out of 10.
So in that sense the competition has already been cheapened, and I must admit I'm at a loss how to rectify this as long as higher-profile, more lucrative European competitions have a bigger hold on the imagination of the increasingly international profile of clubs' ownership and fan base.
Remember all those alleged millions of fans in Asia that Manchester United (for example) like to brag about? They are far more interested in watching their team face Real Madrid or Juventus in Europe than swatting Tranmere aside in the cup. Globalization, TV, and the money sloshing around as a consequence of those phenomena have changed the footballing landscape forever.There is just not as much room in the game as there used to be for the cups.
The same forces that cause some in the English media to claim that the Premier League is the "best league in the world" have also devalued the FA Cup.