Ok. We figured out the issue. It is the DNS setup at the router level. I was using adguard DNS at the router level and Sony Liv app seems to check this (Even for premium users) before playing the Live content. If there's an adblocker, it doesn't play. I have manually changed the DNS on my Android TV to cloudflare dns and it started working.
I was confused with the fact that it was working on my mobile with the same network setup. However, it was working on the mobile because the Private IP setting was ON, which was using it's own random DNS rather than the one setup at the router level. As soon as I turned that setting off, live content stopped working on mobile too, confirming the issue was with the adguard DNS. Below is the screenshot of this setting on iOS.
Hope this helps people who face similar issue.
Special thanks to
@Esmail bro for quickly getting to the bottom of this