MadMonkey: Decent idea, if the Move and Go To, aren't going to work again. (The ones that still aren't)
However, your idea would be even better, and a great compromise, IF the page would stay filtered until you tell it not to.
Right now, if you filter it to show all backgammon games, for example, if you make a move and come back to the main page, it is unfiltered, and you have to refilter it.

And even nicer if "Move and Go To Next game" took you to the next game from that filtered page. (Then no need to go back to main page)

Of course, the only one I'm really missing right now is Move and Go To Next game with non-paying member. Other than that I generally just go in order anyway.