Wow!! That pot in the video was waaaay more than stained! Yuck!
I have a stainless-steel pot also. I picked up a box of polident smokers' denture cleaner tablets (120 tabs was about $5 on amazon)
When it begins to stain, I put 2-3 of them in the pot with hot water and let it sit for maybe an hour. Rinse well and it's good to go.
The same works for stained coffee cups.