on which he is whirled by winds through the air;
In the greek wikipedia he is turning around tied with snakes on a burning wheel in Tartarus, which is a punishment place entered through some caves .
In Perseus one finds the following quote in an article on Apollodorus
According to some, the wheel of Ixion was fiery （Scholiast on Eur. Ph. 1185）; according to the Vatican Mythographer it was entwined with snakes. The fiery aspect of the wheel is supported by vase paintings. From this and other evidence Mr. A. B. Cook argues that the flaming wheel launched through the air is a mythical expression for the Sun, and that Ixion himself “typifies a whole series of human Ixions who in bygone ages were done to death as effete embodiments of the sungod.”
This is some possible symbolism as the sun wheel is found in several mythologies.
Why this punishment? Greek mythology is full of horrible punishments for hubris, see what happened to Prometheus.
In the book Zeus: A Study in Ancient Religion, Volume 3, Part 2, A,B.Cook I screen captured the amphora depicting the torture of Ixion
The light source is definitely sun like, reflections in the water. The wheel is on fire, but not with the strong light of the sun. It could be the artist is thinking "close enough to the sun to burn to a crisp forever". As the modern wish "may you burn in hell" . It is consistent with the Ikaros myth ( the closer to the sun the more heat) .
Looking at the link you provided
Another solar-fire ritual was to roll a flaming wheel down a hill and into a river or lake at the bottom, if there was one. The wheel symbolized the wheel (circle) of the seasons, as well as the sun-disk. The sun as a wheel is an ancient symbol, often depicted in the form of a wheeled chariot being driven across the sky; the swastika is another version of the sun-wheel in motion. In this ritual, the wheel was stuffed with straw or hay (the yellowish color of which resembled the sun) so that the wheel itself was barely visible, and in many cases an axle protruded a meter or so on each side, which people used to guide it down the hill. The idea was for it to roll all the way to the bottom, into the water if any; if it did not roll all the way down, the harvest would be bad
In the lower amphora image, on the right , there is an extension and it looks as if the "nymph"? is sitting on it, and it does look like the axle in the third picture. (On the summer solstice there exists still a pagan ritual with fire in Greece. Fires are lit in the evening of the 24th of June and the people jump through it. It happened in the neighborhoods of Athens when I was growing up).
A possibility exists that the rolling wheel ending up in the water was a ritual in the area , as described above, and Ixion was tied up to the wheel in the myth on a preexisting symbol so that he would burn as a punishment.