I am not familiar with this model, but any cold engine needs priming before it will start, in other words an enriched mixture, especially a two cycle since the fuel has to pass through a cold crankcase before it is transferred to the combustion chamber. That being said, have you verified that the choke is working properly?To reiterate, for a two cycle to be a one pull (from cold) start engine it needs a good shot of fuel into the crankcase from either a few pumps on a primer button or a almost totally closed choke butterfly.

Exactly what Mike said.

Why are you using Premium fuel? It burns cooler and will make more carbon.
The best way to keep 2-cycles running is to be fastidious about air filter maintenance and to use Marvel Mystery Oil in the fuel mix in addition to the regular oil at about 2 oz/gallon. You really can't use too much. When it comes time to store the mower, you can run it very low on fuel, then add lots of MMO. This will make some smoke and serve as a 'fogging oil' for storage.

Don't take this the wrong way, but I do think you are being too picky and anal retentive about the Suzuki. Starting on three pulls is fine by me, and I have owned more than a dozen Suzuki's, including about half which were Commercial Prolines such as the 22040 and 22043.

Yours is an early Consumer model and probably only has a choke. Later models also had a primer button because the Mikuni carb often wouldn't flow enough fuel when cold. However, if you're starting with three pulls, you got no complaints.

Also, You are not doing your engine any favors by running 32:1. Toro and Suzuki have recommended 50:1 since day one (mid 1980's) and oils were not as good 30 years ago. They also approved TCW-3 oils. Before you or other readers go nuts about the TCW-3 recommendation, keep in mind that this is a relatively low revving 2 stroke. It's governed at 3600 rpm and has a max speed of 3900 or 4000 rpm. This is not a high end string trimmer turning 10,000 rpm or a chain saw doing 12,000 rpm. Suzuki knew what they were doing.

Lastly, I disagree with the previous poster about premium fuel. Usually I find him well informed and on point, but not in this case. Premium doesn't run "hotter" or "cooler", just has less tendency to knock and preignite, Main potential advantage to premium is that it is sometimes available without the 10% ethanol. And ethanol is bad news for the rubbers and elastomers in the carburator. Ethanol is also hygroscopic (attracts water) and the resulting moisture will attack aluminum and zamak carburetor parts. If the premium has ethanol, it's then no better or worse than regular gasoline.

My recommendation would be to run ethanol free gas where possible, stay with recommended 50:1 oil premix, and use a high quality oil. A synthetic is great, but really not needed. As for Marvel oil. It won't hurt anything, but I'm not as sold on it as some people. If you start having a lot of trouble starting (8 to 10 pulls), then it may be time to look at the crankshaft seals. These are relatively cheap ($7 to $10) each, but a major job to fully disassemble and replace.

BTW, if you want the Toro/Suzuki manual (492-0260), send me an email, and I'll bounce back with a PDF.

Okay, Hi-octane burns slower, spreading the combustion effect out a little during the power stroke. Thus, peak temps in the head tend to be a little lower on engines that were designed to run regular. This in turn tends to generate more carbon build-up. I actually use mid-grade in my power equipment because the fuel sits out there in the hot shed. When I open it to use and volatiles gas out, I think some of the octane rating goes out with it.