The new Outlook that replaces Windows Mail does not cache mail, so it should fix it. It's basically Outlook on the web in an app framework. No real unintended consequences - but the app doesn't have as many features as classic outlook.
If by upgrading you mean buying a newer version of Outlook - that is unlikely to make a difference.
Try this on the desktop.
Type or paste the following into File Explorer's address bar then press Enter to open File explorer to the folder where the SRS file is stored and delete it.
%appdata%\Microsoft\Outlook
This will reset all of your send/receive settings back to the default.
Go to Send/Receive settings (Ctrl+Alt+S) and change the check every 30 min to 5 min. Do not go lower than 5 min, but higher is ok. See if that makes a different in the desktop version.
That said, its possible there is a server issue - I noticed deletions not syncing today. Read state seemed to.