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The best solution would be for itch.io to provide a cloud saves API. The developers would still have to add support for it though. An OneDrive/iCloud approach would be really difficult to do because different games have different rules about where and how they create save files.

We do want to provide a cloud saves API, but we're not sure when that will happen right now

You could try poking Amazon and/or Google for a Cloud Server.  Game saves are typically under 500 kilobytes in size (usually under 1 MB for most games on here) and the itch devs can even put some limits on the save game file size if needed; make game devs compress their saves before uploading and uncompress them after download (maybe using native 7-zip libraries).