I confess I didn't follow what you meant with the 2x4. Are you talking about doing something to remove the impeller shaft pulley?
For terminology, the impeller shaft is the one coming forward from the engine area, then it gets to the auger gearbox, which then turns the augers (at a lower RPM). I think I know what you meant by auger pulley, but you mean the large pulley that turns the impeller shaft?
Typically, a snowblower will split in half. The bucket assembly (with impeller and augers) unbolting from the engine, transmission, and controls half. Check how the halves join. On my Ariens machines, there are bolts near the top, and the two halves pivot apart, at the bottom. I had an MTD previously, that one might have just bolted together. But take a look first, it might be good to have a helper, who can lower the handlebars gently, to let you split the machine without something crashing down.
But getting better access to the pulley should be reasonably straightforward. Then you'll have to see how that pulley attaches to the impeller shaft. Hopefully it all comes apart easily!
What happened to those electric starter bolts?
One looks like it got worn down by the starter moving around while the mounting bolts were loose. But what about the other one? Did it break off? It looks rather short, with basically 2 threads showing, then the straight section. I would typically expect to see more threads.
You could probably drill out the damaged threads in the engine, and install a Helicoil into each one. But this needs to be done carefully. You don't want to drill too deep, and drill a hole into the crankcase itself, or you could seriously damage the engine.
If it starts easily, and you're not comfortable with Helicoiling the holes, you *could* just leave the electric starter off. With the carb clean, and priming it, my Tecumseh will usually start in 1-2 pulls. Granted, it lives in the garage, so the coldest the engine gets is ~30F. And if it *is* really hard to pull start, the primer hose/tube can sometimes disintegrate, so the primer isn't actually functioning properly. Replacing that broken tube made one of mine much easier to start, since the primer was actually able to do its job.