So using the valet makes it impossible to have run over a nail on the way to the casino?

In OP's mind, yeah.

I valet parked at the Golden Nugget a few years ago and when they went to bring the car out they noticed the tire was flat. They said they wouldn't pay for it since it wasn't their fault, tires go flat all the time. I went up to the garage with the guy to check it out myself. Yeah it was flat and on the top level. I went to Walgreens and got a can of that flat fixer stuff. It didn't work, the wall of the tire was pretty bald so I drove it down to the bottom of the garage and had it towed (by AAA) to my house where I put on the spare. It wasn't their fault it went flat and even if it was there is no way that you could prove it.

OP said the tires are new, ok then if there is a nail in the tread you get it plugged at a tire place (here in Vegas the 2 main tire places don't charge for a plug), if the tire is defective then you go back to the tire place and demand another tire.

As someone who managed a valet operation of a large casino for ten years, I can tell you that how your situation was handled was pretty standard. All valet incidents are documented by a valet supervisor and security (if necessary) and the case is forwarded to the risk management department for resolution.

Flat tires and broken window motors are some of the toughest claims to handle because the vehicle owner is always going to point out that the problem didn't exist before they gave their car to the valet department. And as others have pointed out, you could have easily run over a nail/screw before getting to the casino and have a slow leak in your tire. The best way to handle a flat tire is to offer to change the tire for the customer if they have a spare and offer to take the tire to a service station that performs tire repair if they don't have a spare. Handing you money is not an option, ever!

With respect to Valet parking in general, most large casino valet operations use digital cameras and a computerized parking system to document the condition of the vehicle prior to it being parked. If the vehicle is damaged by the valet attendant during the parking process, the casino is liable for the damages and will pay the claim. If you self park your vehicle, there is a 100% chance that you will pay for your repairs if someone else damages your car. FWIW, I have seen dozens of self parkers, back into another vehicle and proceed to just leave the scene of the accident.

Having said all this, if the Valet Manager or Hotel Manager of the property refuses to give you a $25 food comp as a gesture of good will, then you should probably choose a different casino!

You assume all risks, they are not responsible for anything stolen from your car or damage to it. Read the fine print.

If you bought brand new tires from a reputable store they probably had a warranty. Use it.

Last year I noticed my tire was flat when I parked (self-parking, NOT valet) at the Borgata. I let the staff know and one of the valets fixed it for free right away. I gave him a nice tip and that was that.

IMO the casino should fix it or pay to fix it, irregardless of the BS fine print.

Funny thing...today I got a nail in my tire. I took it to the dealership and had some other work done on the car at the same time. Well, they gave me a loaner, and I drove over to the same casino where you got a nail in your tire. Should I blame them too?