Quick idea which might be flawed:
Let players point the itch.io client to a saves folder (clone) which could a local Dropbox/Gdrive/.. path.
Game(DEV)s (with support) would only specify their local save directory/file per platform and the itch.io client could `rsync` that with the saves folder (a sub folder with the game's id) - at least in Unix based systems that should work out of the box. As an intermediate solution players who know the game's save local folder/file could even specify that per game in the itch.io client themselves.