The only other hotel I've heard of so far with a "no cooler policy" is the Hard Rock.
I will speak for the other side. I love bringing a small cooler and keeping it in my room. I usually store water, beer, booze, and fruit in it. I don't like paying $7 for a bottle of water that is $1 to $2 at the store. I don't like paying $6 or $7 per bottle of beer (plus tip) in the bar down in the casino, when I can buy it for a fraction of that at a store. I don't flaunt it (the cooler), but it lets me spend money elsewhere on a trip. I always take it right to my room. I would never carry a cooler with me while out and about inside the casino.
I have been to the Red Rock, but have never stayed there. They have just given me a good reason to not stay there.
[EDIT] I have an upcoming stay in Vegas in September. One night is at Green Valley Ranch, a hotel/casino owned by the same parent company. The article doesn't mention anything about a ban at other Station casinos. I plan to take a cooler. I will let you know if they stop me.