PrivacyGuides has a section about this on their website: https://www.privacyguides.org/en/health-and-wellness/#menstrual-cycle-tracking
At the time of writing, it seems Drip is the best option on both iOS and Android. The app is free, open-source, and all data is completely local. Honorable mention to Euki as well
Other than Fedora Atomic derivatives (like Bazzite) and of course SteamOS, the only big one I can think of is NixOS? It’s certainly not tailored to gaming, but it does look like 2.4% of ProtonDB users run it
I think you will be relieved at how easy it is. A lot of games are flawless on Deck without the developers even targetting Deck or linux at all, since Proton has gotten so good. Here’s Valve’s guideance: https://steamcommunity.com/groups/steamworks/announcements/detail/3001070242218057029
If you put any stock in IMDb ratings, this may be useful to evaluate each season: https://seriesgraph.com/show/1399-game-of-thrones