What alternative is there, today, that would allow securely doing this without an app store dependency?

Only a few EU countries have rolled out NFC-based eID functionality (as only physical ICAO-based ID verification via NFC is a mandatory part of the EU ID card standard); those are the only ones with a viable path forward in the short term.

The default will likely be the app, but if you have an NFC reader you should be able to use your passport or ID to authenticate as well.

The app has the benefit of being free, getting a working reader costs 60-90 euros last time I checked and Linux driver support isn't great.