I was able to somewhat resolve my issue (for all practical purposes). But in my case, I was anyway moving to a new iPhone.
So I transferred and set up everything on the new iPhone except for Cellular. Then I headed over to the Airtel app and clicked on “upgrade to eSim”. As soon as I did this, there was a pop up within the app which prompted me to “transfer eSIM” to the new device. I tapped on that and followed the steps - these were similar to the steps earlier to activate an eSim but this time around I used the EID of the new iPhone and carefully waited for each step to complete. At the end of all the steps, I was able to get cellular network on the new iPhone (could see the bars show up in the top right). However, per usual, SMS and USSD services were barred for the following 24 hours. These services resumed as usual post that and I’ve been getting connectivity on the new iPhone as you normally would - note that I don’t have any roaming data plans activated.
The old iPhone however, still had the same problem (mentioned above) and didn’t have connectivity. I just erased the old phone (chose the delete eSim option). Everything continues to work fine on the new iPhone!