How does one remove the burnt-on grease spots on a stainless steel gas cooktop without harming the stainless? I have used all types of gentle detergents.
Try white vinegar with 5% or 10% acidity first. Weiman makes some great cleaners for stainless steel you can try too.
When I've burned food into my stainless steel pots the only thing that would get it out was to manually rub them with varying roughness's of stainless steel wool. This of course is going to alter the surface..
I would hit it with a normal dish cleaner. I forgot the brand but the grandma brought over about 6 bottles of a degreaser that would normal be used for washing dishes. A few squirts on my stainless steel oven and the hard grease spots start splitting up. Then I just hit it with a sponge. These degreasers are not green and are somewhat toxic but work and have had no side effects with stainless steel - I know you cannot use oven cleaner.
If you need a gentle abrasive, baking soda shouldn't scratch most finishes. (You might want to try an inconspicuous area first to check that.)
Other than that, and commercial nonscratch cleansers, and perhaps giving the burned area a bath of boiling water to try to dissolve any caramelized sugar or the like which might be helping to glue it in place, the only things I've found that will work without risk to the finish are persistence and "elbow grease" (ie, keep trying and hope you can wear it away a bit at a time.)