I’ve always used Lysol wipes for that. Gently rub until the grime almost gone then finish with a damp MF. The grime keeps the Lysol chemicals from eating away the paint, but the grime is softened with lysol so it’ll come right off.
Steamer just don’t get it too hot. Looks like you handled it well. I love the steamer for contact surfaces like wherever you touch. Steering wheel, shift knob, etc.