I kind of like the fact that your Run knob is also a fuel shutoff. That helps avoid certain problems, like a leaking carb needle valve that lets the gas drain into the engine's crankcase & oil. Though them being integrated together makes it more difficult to shut off the fuel, and then run the carb dry.
At any rate, you're leaving the carb full of gas, if the Run knob also kills the ignition (along with the fuel flow), and shuts down the engine immediately. And when you turn on the Run knob again, the carb would refill with gas, even if it were empty.
I'd speculate that an important difference vs the snowblower is not having a primer on the generator, to help give it some extra gas. You only have the choke available.
Not knowing whether you can add a primer somehow, I'd think that if your carb is clean (doesn't run lean, speed stays consistent, vs surging up/down/up/down), then maybe the "best" approach is just a quick shot of ether before starting it in cold conditions. Or try using some gas in a squirt bottle, perhaps, instead of starting fluid. Not ideal, but if it means 1 pull vs 10, and less risk of causing back trouble, then I'd go with it. "Don't be a hero"