The backlash would be much less severe if there was a credible path to run an open-source implementation that doesn't require handing over highly sensitive data to a third party. The currently discussed solution requires apps to be secured against modification, which obviously violates a core principle of open-source software.
If you want a positive example for a government-sponsored app, look at Germany's CovPass for covid vaccinations, which had independent builds for F-Droid: https://riqq.gitlab.io/fdroid-website/en/packages/de.rki.covpass.app/