You are missing the point.
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Automatic installs from Obsidian do not update the whole framework. So, yes, you need to use a new installer time and again.
See more:
Q&A: Why Obsidian has Two Version Numbers? -
Did you have workspace issues on one desktop? Because the workspace file is different on mobile/tablet anyway. There it is named
workspace-mobile.json. It then will be a different setup, due to architecture.
You can also use different configs for 2 desktops (and mobile as well, although the workspace file will be still different), meaning you rename .obsidian config folder to something else and reload on it. But this is not what you want to do and I think we should stick to the issues at hand.
Did you do what I recommended? Forget the sync thing.