One possibility: if e.g. the dropout or maybe the cage with the jockey and tension wheels... if that thing gets bent a little, then the alignment with the sprockets will change.
I wonder too... maybe as the chain gets worn... the jockey wheel pushes and pulls on the chain laterally. That lateral movement has to get carried up the chain to where it's feeding on to the sprocket. If the chain has more lateral play, then it will be harder to pull off one sprocket. That actually fits what you're seeing. Once that chain gets moving, it can land in the right place. But if the movement is smaller, i.e. just one sprocket, then the chain doesn't want to respond.
Hey, two ideas for the price of one!