I use comet to do the deep cleaning, then a windex wipe after it's scrubbed.

Comet will scratch the stainless surface, won't it?

I've never used comet on my stainless cooktop, but it doesn't scratch my stainless pots that I have used it on. I might be willing to try it in a back corner of the stove, especially if you make sure to only scrub in the same direction as the brushed finished (mine's brushed and looks like it has a 'grain', don't know if yours is).

If there's burnt on gunk in a spot that can be soaked, I'd make a little puddle of hot water on it and sprinkle on some baking soda and let is sit for a while. For other areas, maybe wrap in a wet towel to let it "soak". This should make any caked on areas easy to wipe up.

I have successfully removed some burnt on grease spots near the burners with a baking soda paste - they didn't all come off, but most did.