Jerseys are actually not ideal for cheesemaking because their milk typically has too much butterfat
They're more of a butter cow.
For most cheeses, you want roughly a 1:1 ratio of fat to protein (obviously richer milk is preferred for some blues, bloomies, soft washed rinds, etc, but I'm talking about normal aged cheese - Cheddar, Gouda, Alpines, etc.)
Linebacks should produce great cheese milk. Dual purpose simply means they were bred for both milk and beef; it indicates nothing about milk quality except that those breeds tend to be low/moderate producers.
I think all breeds make great milk and most of the variation comes down to how the cows are fed and how the milk is handled.