You'd think the American custom here is the one that lacks logic, since what does an "entree" grant you entry to? Dessert? Ah, but I have a theory about this. Much like Miss Anne Elk's theory about the brontosaurus, my theory bears no scholarly imprimatur. But it is my theory, which is mine. Ahem...
My theory is that the soup-appetizer-entree-dessert-coffee sequence of restaurant courses is a degraded version of what had originally been ten courses of an American formal restaurant meal: oysters, soup, hors d'oeuvres, fish, entree, roast, sorbet, game, dessert, and coffee. As you lost the specific "roast" and "game" courses, the entree course increased in importance and quantity and became more generalized. It's Darwinian. The hors d'oeuvres and fish courses merged into the appetizer course.
I came upon this theory while I was looking over my copy of The Modern Hostess, which is Volume 1 of the 1907 Consolidated Library of Modern Cooking and Household Recipes. This volume begins with a "formal hotel dinner" as illustrated by Oscar of the Waldorf (talk about your scholarly imprimatur!). For each course there is a description and an illustration, including an accounting of the proper tableware. All I include here is a description of the food. If you want to know about the tableware and its placement, just ask.
The First Course: Huîtres -- Oysters: The oysters should either be thoroughly chilled, or served on cracked ice, and accompanied with a bit of lemon. Horseradish and pepper sauce are the proper articles to serve with them.
The Second Course: Potage -- Soup
The Third Course: Hors D'oeuvres -- Dainty Dish: may consist of cold side dishes, such as olives, radishes, canapés, caviare, anchovies, etc., or timbales, croustades, palmettes, mousselines, bouchées, etc., which are served hot.
The Fourth Course: Poisson -- Fish: The fish, if it is boiled or fried, should have potatoes served with it; if broiled or cooked in any fancy manner, serve cucumber salad.
The Fifth Course: Entrée: is in this case sweetbread with green peas. Entrées should be made as light as possible, and they should be made in a fancy way, so as to avoid any carving.
The Sixth Course: Rôti -- Roast: consists of the "pièce de résistance," served on a single hot plate. Saddles of either veal, mutton, lamb, venison, or antelope may be used for this course; also turkey, goose, duck, capon, etc., may be served, accompanied by one or two vegetables.
The Seventh Course: Sorbet -- Punch: which is not essential, but nevertheless a pleasing half-way house among the various stages of the dinner, is sherbet.
The Eighth Course: Gibier -- Game: serve with salad, follows immedately after the sherbet.
The Ninth Course: Dessert: introduces the dessert, which, of course, is served according to the menu.
The Tenth Course: Café -- Coffee consists simply of black coffee in small after-dinner cups; cheese, if served, accompanies this course.
The "Roast" is the course that corresponds to an American entree, except that not all modern main courses are roasted so that name didn't work. How did salad get moved from being served with the Game course to being its own course at the head of the menu? I guess you had to put it somewhere once game became less plentiful.
The origins of "entree" as "main course" must lie somewhere in this overwhelming gluttonous dinner standard of a hundred years ago.
Now I have a strong craving for roasted antelope.