Salary cap without tweaking or overhauling transfers wouldn't do anything in relation clubs spending high amounts in salary and transfers.
With salary cap, what doesn't get spent in salary will get spent in transfers, raising transfer fees more as you alluded, with that being the problem then.
Clubs/directors have one budget for salary+transfers, not separate.
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Also, how do you impose same salary cap on EPL clubs when they're earning 5x more?
North American system really is the way to go.