I just deployed the same files built with 'flutter build web' as the Azure Static Web App and the workaround below worked.

 assets:
      - packages/timezone/data/latest_10y.tzf

Load the timezone database with:

await tz.initializeTimeZone('assets/packages/timezone/data/latest_10y.tzf');

This workaround did not work when the same files were deployed as Azure App Service. I am still interested to know if there is a way to make it work on the App Service, in case someone figured it out.