I'm thinking of an idea right now.....
OP, or whoever wants to head it up, chooses the game each week. He/she can choose a game that's available on the sub services, or not. But, it should be a fairly low cost game in any case. Then, anyone that wants to participate can get the game in whatever way and on whatever system they want. Maybe when we get to sale season games could be selected for a few weeks in advance, so that people can pick them up on sale.
This could still favor games available on the subscription services, but wouldn't rule out everyone else.
I tried to do this with XBOX live content service and Game Room. While I had decent success with Gane Room, to go with any low cost game in there did not work. Also, having winner pick and setting terms if play was an incentive to keep playing that did not require buy in and paying out of money. The reason to switch to a new game was to cause an eventual change in games.
If one were to do a head to head fighting game challenge to cause players to pick a fighting game, one could have the defending champ keep picking the same fighting game until they got defeated.