We use a travel chair for my son, and we either gate check it or check it as baggage depending on connection times. However, regardless of how we check it, we always take the feet off and put them in the overhead bin. His travel chair gets damaged no matter which way we check it, but at least we don't lose the feet. Gorilla tape is our friend.

Also, I have never seen the storage closet on board as being large enough to store a wheelchair or even something as small as my son's travel chair when it is folded up.