Good discussion yesterday as I pulled out the first concrete question, and it made me realize a few other critical points that merit mention about this approach.
Your Score Is Not Your Fun - The Apgar score is not a direct measure of the child's health, it's just in the ballpark. Similarly, this score, whatever it will be, is not a measure of your fun. Just as a kid with a low Apgar score can turn out just fine, you might have a game with a terrible score that totally rocked. You might decide that means the score is meaningless, but to me, I'd look at it as a reason to ask "Well, these things didn't work, so what _did_ work so well to make it awesome?"
No One Gets a 10 - One other useful datapoint about the Apgar score is that it's really rare that a child get a 10 (or more accurately, a 10/10). Something is usually off, but that's expected. The same is going to be true of every game. Someone's going to be tired or distracted or having some kind of problems. The goal is nto to get a 10, it's to capture the state of things. The score is not the end point, but a midpoint measurement, and that's important for the next point.
We're Not Measuring One Game - The Apgar score is a handy shorthand in the delivery room, but it's real value is as a datapoint in a much larger dataset. If it's high, then it's probably not a big deal, but if it's low, then what? How do things go for the kid? From a hospital's perspective, if they want to improve, the score gives them a metric to shoot for, so that more kids with lower scores do better. The same is true of your game. Looking at your games over time and seeing what changes, what correlates with good and bad games, and thinking about that is going to be much more useful than rating one game and drawing big lessons from it. This is especially true if one metric doesn't seem to apply to your group - if it always comes in low, but you still have fun, then the score is not a criticism, it's just where your baseline is.
Who's Responsible for What? - Even though the GM is not solely responsible for everyone's fun, she would be well served to act as if she was. This is absolutely a personal bias, but I want to lay it out on the table. For the purposes of asking questions, I am assuming the GM is taking responsibility for everything, even though things will often be out of her control. This may seem unreasonable, but I want to point back at the rest of this post. If you end up giving a game a low score because one player was so hungover that it just dragged the whole session down, that's not your fault, but it still made for a bad session. The low score is not a criticism of the GM, it's a report of what happened. The hope is that it's useful data, not a judgement.
Tomorrow, I'll see about extracting a second yardstick.