The other thing is that this implementation doesn't seem to support overlapping tiles, which is kind of important. For instance, the topmost tile should overlap and block all four tiles under it.
No.
According to wikipedia[1]: "A tile is said to be open or exposed if it can be moved either left or right without disturbing other tiles.". Also look at the photo in the wikipedia article[2]. This implementation looks correct to me.
[1]: https://en.wikipedia.org/wiki/Mahjong_solitaire
[2]: https://en.wikipedia.org/wiki/Mahjong_solitaire#/media/File:...