I think there’s a myth that players don’t get picked if they don’t play for the big 6.
The current squad includes:
Pickford (Everton)
Butland (Stoke)
Hart (West Ham)
Maguire (Leicester)
Bertrand (Southampton)
Keane (Everton)
Cork (Burnley)
Livermore (West Brom)
Vardy (Leicester)
Who do not play for the big 6 and Abraham and Loftus Cheek who, whilst contracted to Chelsea are playing their football elsewhere. There have also been callups in the fairly recent past for the likes of Chalobah (Watford), Gibson (Middlesborough), Forster (Southampton) etc.
Playing champions league football improves your chances as you get to pit yourself against top European teams. It’s also reasonable to assume that usually, the best players will gravitate towards the best teams so there’ll be a large percentage from the big 6 but it is entirely possible to play for England without playing for one of the big 6.