Good news now.
Finally I was able resolve the issue. And issues was totally unexpected
Static Charge on the remote.
In old days when Laptops suddenly use to stop Powering ON, we use to remove power Cord, remove Battery and press, hold Power button for 40-50 secs to remove any static charge. And it use to work 60-70 percent of the time.
Same thing just clicked and I removed the batteries from Remote, pressed power button for almost a minute and then left it without batteries for 4-5 mins. And it worked (Previously I tried changing batteries with new one but it would have been done within 10 secs)
Then I was able to see the IR LED, and remote discovered iin Bluetooth in my phone when I put it in pairing mode.
Now works fine with Stick
However, I am keeping my replacement request open, anyway it will take 2-3 days to approve and in the meantime, I would find if the issue is repeating
Thanks very much everybody for their help!