Using Classic Outlook on Windows for personal email, calendar, and contact management
This workaround worked for me.
I unistalled the new Teams and clossed Outlook classic desktop app.
Then I installed Teams classic from the official download page, clicking the button to download for personal accounts.
Next, I logged in with my work account. This step made the trick for me, otherwise the add-in does not appear.
After restarting Outlook, the Teams add-in reappeared and worked well.
Finally, I reinstalled the new Teams from the official web page, clicking the button for work and school accounts, and logged in with my work account.
Until now the Teams add-in is working as expected. No need to install the new Outlook (which btw looks horrible)