I don't think they have put a 10hp Briggs on a Snapper mower in 30 plus years. That engine is NOT a candidate for synthetic oil. HD30 is what I would run in the old Snapper for sure!
Other engine, if newer and in good shape would get the same HD30, Rotella/Delo 10W30, or any synthetic 10W30 if it were mine, considering your climate.