Does the FA & England Need To Regroup?

Following England’s exit from the Euro 2016 tournament at the hands of Iceland, does the FA and the  England Manager need to rethink what league they select players from to play for England?

At the moment all the players who play for England play in the English Premier League, execpt Joe Hart who plays in Seria A in Italy for Torino, on loan from Manchester City.  Only Tom Heaton played in The Championship for Burnley, who now play in the Premier league.
My question is this.  Should the England manager select players from lower leagues such as the championship? The Premier League is the richest league in the world and attracts many foreign players from all corners of the globe, meaning that the pool of English born players is getting ever smaller.

Other home nations Scotland,Wales & Northern Ireland get their players from the Premier league, The Championship and the Scottish Premier League. England only get players from the Premier league.
England are becoming a national joke and we are hated within the ranks of Uefa and FIFA, who themselves are not whiter than white, in my opinion Sepp Blatter and Michael Platini should be be serving time for corruption.