If nobody ever Finalizes, then the deadlines will remain consistent between the games, although they will shift 5 or 10 minutes later each phase because the php program gives that grace period. But that should be OK.
For metagaming, it may be simpler just to accept that, OK, we're all playing the same 3 games with the same 7 people at the same time, so let's metagame between those 3 and just make that part of the rules. Which doesn't mean that you can't ally with a player in one game and fight against that same player in another game! (I played a game once where I was allied with a power on the west side of Switzerland although we were enemies to the east, and it didn't confuse us. In WWII, the western Allies and the USSR were allied against Germany, although the west was neutral in the war between the USSR and Finland while the USSR was neutral in the war between the Allies and Japan.) But this would validate threats, promises, reprisals, and rewards across the 3 games.
Like youradhere, I'd also like to wait until my current game finishes. I want to play only one game at a time to avoid burning out, although I would consider this set of 3 as one game. If you do it without us, let us know how it goes!