If PL especially brought in a wage cap a lot of the foreign talent would prob choose to go elsewhere. That would lose some of the international market interest (which is PL main growth area) due to not having players from that nation playing here and longer term a lower standard. That would result in lower broadcast bids, reduced income, thus players would have to be paid less, less foreign players come and our better players going abroad for better money and so on in a downward spiral.
The fact that they run on such high costs in terms of player wages means the entire set-up is madness and if the broadcast deals did ever go down in price many clubs even at that level would be absolutely screwed.
Championship could well be the same but it would stop teams massively overspending to get promoted because that massive financial reward wouldn't be there (or at least massively reduced)
