Unable to update Apple ID
I spoke to the customer support again. This time an attentive guy there was able to point out the issue that is happening with my account.

Apparently Apple has put some restriction on new iCloud email accounts and they cannot be used as primary Apple ID for first 30 days.

They have added the following note in the documentation page as well but it is written in fine text so it goes unnoticed. One has to pay attention to notice it. It says:

"If you try to change your Apple ID to an @icloud.com email address created within the last 30 days, you might be asked to try again later."

I made my @icloud address 2 days back, so I guess in my case I have wait for 28 more days after which I hope I will be able to change my primary Apple ID address. Let's see.