I've used 10W30, 5W30 conventional oil for twenty years in two Tecumseh Snow King engines. The engines were still running like new and outlasted the machines they were on. Brand of oil didn't matter as long as it was of decent quality, Valvoline, QS, Shell, Texaco, Castrol, etc. A 5W40 or 10W40 oil will work just as well.
I presently have a B&S Kool Bore (aluminum bore) engine that I had run nothing but 5W30 Mobil 1 synthetic in it for three years. For some reason, that engine consumed the Mobil 1 at a rate of about 1/2 oz. per hour. Way too much consumption in my view. The old tecumseh engines never used a drop of conventional oi over the entire season.
This season, I have switched back to conventional oil, Castrol High Mileage 5W30, to see if it is a better choice for this B&S engine.