"This plane cannot take off out of there fully loaded because at that elevation the tires are not able to spin fast enough for the airplane to build up enough speed.'
I don't think the elevation makes the tires spin slower.
Even if it did, the tires don't push the plane down the runway, the engines do. Or maybe I misunderstood what you were trying to say.
Two factors come into play in determining how much power a jet engine will produce: a)Temperature of the air and, b) Altitude of the air.
Warmer air is less dense air. Less dense air means the engines are getting less material to produce thrust.
Likewise, higher altitude air is also less dense. Again less material for the engines to produce power.
Yes, it is true, higher altitudes produce cooler air, but density is more sensitive to altitude than to temperature. The atmospheric temperature decreases only by 3 degrees F per 1000 feet. And even Denver gets red hot in the summer, its elevation notwithstanding.
Often a plane like a 747 will have to decrease its load to affect a safe take-off on a very hot day anywhere in the world. If it is also a higher altitude airport, even more so. Denver on a summer day would be a double-whammy for a laboring 747 full-loaded like a boat leaving for China.
The engines can produce enough power for flight of course, but with less dense air they need more time to build up to that. That means they need a longer runway.