We normally use left over rib roast for cheesesteaks but I have used New York steaks a time or 2 when they were on sale.
Not as good as rib roast of course, but not bad.
I cook it just like you said, chopped up into small pieces and cooked at medium heat in a bath of au jus.
Peppers, onions, and wiz on a roll.

Will give that shot next time as I love Philly Cheese Steak sandwiches.
I also used a few slices and had steak & eggs one morning.  The eggs and thin slices get done at the same time.  I used two separate frying pans.

I have seen where Costco also has thin sliced Rib Eye from time to time.  Probable would work better for the Philly Cheesesteak.  Will keep my eye open for it.
For the fajitas that I made I cut the pieces you see in the fry pan above into long strips.

@ZCZ -I was thinking the same thing about a cheese steak.  Never heard of anyone using a NY strip for it, but I think it would work.  IMO, you need to chop it up good while cooking it and probably setting your stove top at about medium while doing so.  I am interested to see one of you give it a try and provide your thoughts on your outcome.

Maybe call it a New York cheese steak....If I gitrdun I will.....