It turns out, the problem solved without much interaction.
The 1st installation, I waited for around 1-2 days and the status never changed. But in the 2nd time, a couple of hours were enough. Traffic is flowing correctly and it is listing a healthy node in Azure. Although, Bird shows the networks announced via BGP as unreachable, it's actually reaching and deploying the cache content to our selected customers.
The documentation is indeed a little bit uninformative and lacks some in-depth troubleshooting methods. Anyways, everything is fine right now.