yichusdik, the halacha is that you are not allowed to play a game on Shabbos that normally involves writing. This is the halacha, so it doesn’t matter what we think. But, it’s not comparable to a hand. The hand’s function is not writing; that is merely one thing it does. If you’re playing a game that doesn’t normally involve writing but you happened to have once or twice written while you played it, I would imagine that it’s not assur to play it on Shabbos.
btw, what we used to do when we were kids: even when we played Scrabble during the week, we would play w/o writing so that it would be muttar to play on Shabbos. We took a catalog or big book and turned the pages to keep track of the score. I’m not paskening that this makes it muttar to play on Shabbos (ask your own Rav) – I just remember that’s what we did as kids.