Hi Randy, my issue has been solved here's a detailed follow up
So I tried every, everything! to get that audio output back;
1)I tried installing older windows version, no luck
2)I tried hard resets, driver delete and reinstall and even installing generic drivers from motherboard manufacturer. No Luck
3)I searched the internet a lot, and by a lot I mean hours of scrolling forums, and driver pages but still, no luck
PC Specs:
Intel H61S motherboard
Windows10 22H2 installed on GPT partition table (not MBR)
Aptio setup utility BIOS (UEFI supported as well as active) by American Megatrends
Till now I had followed all the steps we discussed in order, twice but there was no response.Then I found some random forums which said BIOS settings might sometimes interfere in HDMI outputs, with minimal hopes I checked my BIOS.
I went on the path: BIOS→Chipset→PCH-IO Config→PCH Azalia config
There I found the word HDMI mentioned which was enough for me to understand that this must be enabled, initially it was set to auto, I manually enabled all the parameters and my AUDIO WAS BACK IN NO TIME
I am glad that across the internet this (no HDMI audio output to TV) is a very common issue (apparently) but no forum had a mention of Azalia configuration as a solution to this problem specifically.
I'd like to share the screenshot as well for future references as well as other users benefit.
Thanks Randy ! Your efforts helped me.