Can I paint cooktop with BBQ paint? Or perhaps enamel paint?

I wouldn't bother with painting stuff that has survived, in sound condition, being unpainted for 40ish years to help preserve it for another few. Painting galvanized stuff is not always the simplest to do reliably.

For ease of cleaning, maybe just put a layer of aluminum foil down in the bottom of the tray. When it's too soiled, pull it out and replace it.