I would agree with that assumption. However, with a new carb, new lines and fresh gas it would seem almost impossible that fuel is the issue. I did make up a small fuel tank to try if there is potentially something wrong internally with the tank on the snow blower.
 
As far as mechanical items, does everything look ok in the 2nd video?

The kill wire is hanging right now and disconnected from the shroud piece where the plastic key goes. So, it's definitely not that.
 
I'm not sure if I mentioned it earlier. While the engine was running. I tried spraying carb cleaner in very small amounts increasingly more and more to test if it was a fuel issue. It made no difference in how things are running. What this engine reminds me of is when you have the distributor installed wrong in a larger engine. Acting almost like the timing is say 30 degrees off. Just enough to barely run.
 
I'll order some gaskets then throw the engine back together as it seems to be perfectly fine inside. Then move on to checking the ignition.

Found the problem.
 
The flywheel key is sheared. I just picked a new one up but have to wait on the gasket kit to arrive. I'll post another video once it's running correctly.
 
https://www.youtube.com/watch?v=DMi8WT-sngU

AVB
Glad to hear you found the problem. Usually these engines will not run with a sheared key.
 
Just remember to torque the flywheel to 65 lb/ft (88.0 Nm) or it will shear the key again quickly.

It barely ran if you watched in the earlier video. I should have checked the key way earlier as I knew something was way off. The muffler got so much hotter than it should have. Which I now know is because the ignition was retarded by 30 degrees or so.