Adding to sleepyTonia’s comment, many flash games have been preserved through Flashpoint Archive, which is like an epic DRM-free Steam client for flash games (as well as other web game technologies, like the shockwave player). However, Flashpoint uses old flash player binaries that, as stated, may one day stop working as hardware and operating systems evolve. If that happens, it’ll be great to have a replacement interpreter ready to go that can be compiled to run on newer tech.
A non-steam game can be launched through Steam on either device, but Steam doesn’t sync game saves for non-Steam games, hence Toribor’s use of syncthing. Once a sync job is set up for each game’s save folder, it’ll keep them synced about as well as Steam does for native games.