I confirmed it (your issue) on my system. Everything works without internet, as long as your phone/tablet was already logged in. But an actual internet link, on the phone at least, needs to be available to log into the App, even with Local Server.
I suspect this is meant to be this way, and not a valid Issues and Errors situation (EDIT - I reverted back to this as per Pavel’s post here … possibly just Blynk protecting it’s closed source App and commercial operations?