That looks like only surface rust to me.
My solution has been to carefully, but thoroughly, sand the affected areas with high grit sandpaper and then paint it over with Rustoleum black grill paint. It certainly works. It is made for this specific purpose. However, the paint dries to a dull, flat, not-shiny black, so you can definitely tell where the touch up has been done if you are standing within 4 feet of the grill. Farther away, it's not a big deal.
I would love to hear if anyone has a better suggestion to match the paint, or the shine, in a better fashion.