I know that likely he was left behind for real-life actor leaving reasons.
But I think it made sense story-wise as well for Mat as a character to a certain extend.
He has always run away from being a hero and avoided responsibility. Maybe he'd feel sorry for leaving them in the second season...
This is Aes Sedai. They tend to keep cool angreal and ter'angreal they have secret. Probably it's just a secret between Siuan and Moiraine.
Also Siuan had a similar painting in her room so I suspect it creates passage only between those two. So it cannot be mass-produced. The writers can make...