are the saves stable or can you lose progress if you don't play the whole thing?
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Unless you clear your browser data or have incognito mode enabled, saves should persist even if you close your internet browser and shut down your computer.
If I release an update for the game, it might (but not always) disrupt saves. That said, the next projected update is in early February, so it should be safe until then.
It's a private mode on your web browser. But it is probably not enabled on your web browser so that shouldn't be an issue.
https://en.wikipedia.org/wiki/Private_browsing
Are your saves working? If yes, then incognito mode shouldn't be an issue.