Duh! About 5 Deg F per 1000 feet as I recall. (I used to be a pilot.) Temperature isn't the problem for grass. There's plenty of grass in Florida. The problem for all vegetation is that at some point there just isn't enough CO2 in the atmosphere for it to grow, Al Gore notwithstanding.
The writer of this article said altitude wasn’t diagnostic, which is what you now agree with, but nearly immediately had said was BS.
There is no grass in Antarctica, or above a meandering line along the Arctic Ocean. So, yes, temperature does have an impact. On steep heights the problem is generally, no soil, and the higher one goes, lack of liquid water. There’s nowhere on Earth (above water at least) where the CO2 level falls below where plants can survive.