I did an interesting experiment today, I have a container of "Vanish" stain remover, for when I get bad stains on anything, there was just a small amount left, so I decided to try it in my stainless steel sink.
My sink does get turned brown with all the tea pot emptying's, and the bottom of my bowl also gets brown for the same reason.
I clean it now and then the hard way, detergent and a scrubbing brush on the bowl, and a stainless steel cleaner on the sink.
Today I half filed the sink, tipped in the vanish, stood the bowl in, put some water in the bowl to stop it floating, and forgot it for an hour.
When I check the bowl was just like new, and so was the sink, I will certainly be cleaning it that way again, no effort at all.