Yes, on small engines most are still flat tappet because of the cost of rollers and the absence of CAFE rules. You are correct that an API rating with "Sx" indicates gasoline suitability ("S" for "service", not "spark" as is commonly held).OK, Interesting info. LS JR. did not point out this nuance. Do you know which oils are dual rated? It has an API rating on the package starting with an "S"? Not to extropolate too much here, but I am going to assume all of these guys on tractor and small engine forums suggesting everyone to use a diesel oil in their small air cooled gas engines do not know the difference you are pointing out.
I can see how that type of oil appeals to a fleet manager. They can inventory fewer oils.
Keep in mind this is a small air cooled engine. I would be willing to bet it has flat tappets. Kohler does use roller followers on larger water cooled engines. You may be able buy aftermarket roller follower kits for these engines. This is what the racing guys do. I doubt it was speced out on this Van Air application. But I do not know for sure..
At this point, I think the OP has his answers.
Chevron Delo, Valvoline Premium Blue, and some grades of Delvac (1300 super) are all dual rated for gasoline and diesel. You can even get full synthetic 15w-40s that are dual rated now.
The gasoline-rated HD oils are a hidden gem because the higher viscosity gets an exemption from many of the neutering effects of ILSAC and Energy Conserving designations. So an API rated 15w-40 or 5w-40 is almost always a better oil by additive package than a 5w-30 or 5w-20 in the same or similar product line. This is because the formulator isn't obliged to honor arbitrary rules like capping Phosphorus at 800ppm; they can set it wherever their own testing deems it appropriate.
Valvoline's full synthetic 15w-40 Heavy Duty oil carries all the usual and expected diesel certifications, but it also meets API SN and has a very stout full-SAPS additive package with 1100ppm phosphorus and 1240ppm zinc. The 5w-40 full synthetic Premium Blue also has a similarly stout, full-SAPS additive package.
The Delo 400 XSP is a similarly excellent 15w-40 with very stout full SAPS additive package (1100ppm phosporus, 1200ppm zinc) and also meets API SN Plus.
Finally, Delvac 1300 Super in 15w-40 meets API SP. It's a 0.9% SAPS oil that, while only a syn blend, is an excellent overall choice and is probably the champion of performance-per-dollar on normal drain intervals. It's excellent but cheap enough you can drain it without feeling guilty.