First, if you banned the larger teams from the FA cup then nobody would sponsor it, and the FA need the sponsorship money.
Secondly, Liverpool are in the Carling Cup semi final (two legs) AND the FA cup and have just played a tight schedule of matches over Christmas.
They played on the 26th December, 30th December, 2nd January, 5th January, 8th January and are still due to play on the 13th of January and the 17th of January and 23rd January.
No players or sportsmen can expect to play every match of a schedule like that.
Thirdly, they have a terrible injury list, with 11 first team players out injured (Henderson, Sturridge, Gomez, Lovren, Skirtl, Ibe, Coutinho, Toure, Saakoh, Origi, and Ings). They have no fit first team centre halves.
If you are going to blame anybody blame the football authorities (all of them) who expect players to play almost continuously.
Every other year we have either the World cup or Euros which takes up most of the summer. They have now decided to have 32 teams in this years Euros, stupidity.
They are also looking to increase the number of teams in the World Cup.
Also teams like Liverpool, Chelsea, Man U, Man C etc go off on long tours to the USA or Asia so the players never get a break.
I run a number of Fantasy football teams and I have never had to make so many transfers due to players being injured. All the top teams have had players out (Kompany, Aguaro, Wilshire, Welbeck, Courtios, Valencia, Shaw and many others have all had long injuries).
Football really needs to take a long hard look at itself.
My first suggestion would be to reduce the Premier League to 18 teams (4 less matches a year) and set up a Premier League B (second tier) to replace the Championship. That would be a start.