How to clean the stainless steel burner "bowl"

by Andrew
(Santa Barbara CA)

Hi - I've been trying to keep the 9" by 9" stainless steel burner "bowls" which surround the gas jets on our stove clean.

A previous cleaner used something too abrasive and some parts of the metal got scarred. The stove is quite old so we cannot get replacements parts.
I used MAAS cleaner for the last few months, the parts not only were clean but generally shined and looked awesome. But it seems now the burners show more spots and almost have a light greasy residue and they seem to require more cleaning than ever.
Can anyone recommend a product you've found that should work for stainless steel that is also exposed to some cooking heat indirectly? Thank you!