I spent some time delivering and making pizzas. Not for a Ray's, but because I learned how to make pizza, I became better at figuring out what places used for sauces, topping methods, and the cheese mixture.
Pure mozz is very distinct in texture and grease level (not very present). Adding even a quarter or a third white cheddar raises the grease level up a lot. It changes the texture. Most people would see it as different than a pie with just mozzerella on it, but wouldn't think that the difference would be cheddar. It doesn't taste like cheddar. It doesn't look "cheddary". People won't even notice most times unless there is more than half cheddar or if you use orange cheddar.
Pizzas can be oily because of toppings, oil added to the base or to the sauce, but most of the time it's because an oilier cheese is used, usually in a mixture. Since Cheddar is usually more economical, it's usually used.
You don't have to take my word for it. Make your own pizza with a bit of medium white or mild cheddar mixed in (1/4-1/3). I bet you will find the cheese tastes about the same (at least cheese wise) as most NY slices and is about as greasy.