I wouldn’t like the melted on top idea. A piece of nice cheese on the side… That I could go for, once in a while. The American cheese idea - forget it. (It’s often good on a burger though.)
Apple pie without cheese is like a hug without a squeeze.
New Englander here. Never tried it melted. Also salt my watermelon.
According to this article from National Geographic, butter from corn-fed cows is naturally white, while butter from grass-fed cows is naturally yellow.
I lived in Illinois in the 1960s and we used to take family trips to Wisconsin. I remember big signs at shops just across the border in Illinois announcing the availability of colored oleomargarine. The NatGeo article confirms that yellow margarine was illegal in Wisconsin until 1967.
Cheese is routinely put on just about anything, as I recently noted in another thread, and much of that cheese is melted, so no surprise it would be put on apple pie.
Which doesn’t mean it’s a GOOD idea. Far from it. Way to ruin a good slice of pie!
Having said that, I’ll admit that I’ve never seen melted cheese on apple pie; even though dairy farming is pretty big in this area and apples are a huge crop hereabouts, it doesn’t seem to have occurred to anyone around this area, or maybe they realized it was a fairly disgusting idea.
When I was in fifth or sixth grade, my class went a field trip to a dairy plant. The guy in charge of the butter section told us about how color was added because people expected butter to look like that. He also said it varied a lot across the United States, from nearly white in some parts to almost orange in others.
As for apple pie with melted cheese, it doesn’t sound very appetizing to me. Hot apple pie topped with vanilla ice cream, yes. But with apple pie and cheese, they should both be cold.
Hot apple pie topped with a dollop of cold whipped cream I could go for. Also hot pecan pie topped with a spoonful of cold butter. Mmmmmmmmmmmmmmm! :o
Same here – I tried it as a youngster because I saw it on TV. Meh.
I’ve had apple pie with an unmelted cheese substance on top in restaurants back east. It must not have made a big impression on me either way. I never eat apple pie that way now. My wife likes ice cream or whipped topping on her pie. I can take it or leave it.
I occasionally will slice up an apple with slices of cheddar, (sharp if we have it), to go with for an evening snack though. 
I have made apple pie and pear pie with cheese baked into the crust.
I was gonna say cheddar, so sharp the crystals crunch when you bite into it, but just put beside or on top, not melted.
Likewise, I’d heard about it a couplea times, and I said, “Better try it, or I’ll wonder forever.”
The result was, as quoted, more or less – Meh. It was tasty, but the pie was tasty enough, since I use modern apples, sweeteners and spices, so it didn’t add much. I did it on Thanksgiving, in front of everyone, and everyone thought I was strange. Now, thin shavings of Parmesan, on stewed or baked pears, now that makes sense to everyone, ethnically.
Pie isn’t pie without Cool W-hip. 
Pie without cheese is like a kiss without a squeeze! Never heard of melting it though.
Heard of it, never seen it or tried it.
I would try it, but it doesn’t sound appealing to me.
My wife likes to mix up some cinnamon into honey and dip apple slices in it.
The green chili apple pie in this thread has a cheddar cheese crust that I found to be pretty tasty: Apple Green-Chili Pie With Cheddar Crust and Walnut Streusel - Cafe Society - Straight Dope Message Board
Re: simply putting cheese on the pie, it’s one of those things people say “I’ve heard that some people…” but I don’t think I’ve ever seen it. Might have tried it but I don’t remember. It seems like it would be tasty but I never think of it.
I once read an essay that explained what a true Yankee is, and it boiled down to “being from Vermont and eating cheddar cheese with apple pie”. I happened to mention that to my dad once, and he said, “My father was from Vermont and ate cheddar cheese with apple pie.” So I’m the granddaughter of a true Yankee. (In his case, the cheese wasn’t melted on it, it was just on the side.)
I don’t put chees on my apple pie; but a good slice/wedge with my apple pie is great. I like a good, sharp cheddar.
Not even just one little chee? 
Lucy Ricardo once tried to tempt Fred Mertz with a slice of apple pie with cheddar cheese; I assume they were both native New Yorkers.
My mother also used to say “Apple pie without cheese is like a kiss without a squeeze,” but she NEVER served them together (NE Ohio).
If a variety of pie is available, my choice is usually blueberry or lemon meringue or Key lime or cherry or some sort of cream pie…apple would be VERY low on the list. While the idea of apple pie with cheddar sounds agreeable, I may never taste it.
I do it, I prefer a slice of colby cheese.
Supposedly it is a Vermont thing. Not that I’m from Vermont, but thats supposedly where it started.