Changing the name of the league won't make any difference. And as for PL money trickling down.........the clue is in the name: PL money. It's their's, and they ain't going to let go a penny of it if they can help it; after all, they need every penny they can get to pay £100K or more a week to...
Top sides in any league do try and play football more than the teams at the bottom of any league, who will just be scrabbling around to avoid relegation.