For me, American cakes aren’t greasy. They are generally pretty moist because they contain a lot of fat and sugar (both of which help to inhibit gluten formation). European cakes tend to be fairly dry because they contain relatively little sugar and fat, allowing for a dry but airy structure perfect for soaking up syrups without becoming soggy. Since American cakes are fairly-to-very moist to begin with, I rarely add a syrup as the more tender crumb can get soggy and mushy. And that’s just gross. So, to answer the question, it’s not so much the amount of protein in the flour (although that is a factor) as it is the addition of fat and sugar that contribute to the more tender crumb of an American-style cake.