GSM goes offline

Although this sounds like a logical conclusion, it’s not.
Your phone uses the SIM for different types of communication, even when running the Blynk app, the Blynk communication uses different ports and protocols to an Arduino with a GSM shield.
The topic below is about a different scenario, but demonstrates the issue:

Yes, with debug turned on (as you have in your code) the next line you should see after your auth code is sent to the server is:
[timestamp] >[00|00|01|00|C8]

This means “Blynk success”
See here for more info:

As you’re not seeing this, it means that the connection request from the Arduino isn’t being seen by the server, or the response from the server isn’t being seen by the Arduino. My guess is the first, that the server isn’t seeing the connection request at all.

I think you should try some different SIM cards. Try begging, borrowing or stealing them from friends, family or random strangers in the street until you’ve either found one that works, or convinced yourself that this isn’t the issue.
I think that there’s little point in trying multiple SIMS from the same company, or from a subsidiary company. This table on Wikipedia shows the parent companies that you should aim to try, and you’ve already tried the first two…

Pete.