Except that mid-table premier teams generally field weakened sides and don't care if they lose cup matches so that they can concentrate on survival and safety from relegation.
Really? FA Cup Final lineups since 1992:
1992–93 Arsenal v Sheffield Wednesday
1993–94 Manchester United v Chelsea
1994–95 Everton v Manchester United
1995–96 Manchester United v Liverpool
1996–97 Chelsea v Middlesbrough
1997–98 Arsenal v Newcastle United
1998–99 Manchester United v Newcastle United
1999–2000 Chelsea v Aston Villa
2000–01 Liverpool v Arsenal
2001–02 Arsenal v Chelsea
2002–03 Arsenal v Southampton
2003–04 Manchester United v Millwall
2004–05 Arsenal v Manchester United
2005–06 Liverpool v West Ham United
2006–07 Chelsea v Manchester United
2007–08 Portsmouth v Cardiff City
2008–09 Chelsea v Everton
2009–10 Chelsea v Portsmouth
2010–11 Manchester City v Stoke City
2011–12 Chelsea v Liverpool