It took me a whole day troubleshooting this problem with my PC and using a clean VM to compare folder and registry state and finally managed to fix this issue.
To make the sync option available on Edge open registry editor and delete the key HKCU\SOFTWARE\Microsoft\Windows\CurrentVersion\SettingSync\SyncPolicy
A clean install of Windows 10 doesn't have this key so it must be a leftover from upgrading that's not used anymore.
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