It's a 2 cycle so no oil.
The plug gets flooded real bad. I pulled it a few times to wipe it down and heat it up. I did have a problem like that a while ago. I had to tear apart the carb and clean the screen in the bowl area. Real tedious crap... So, if the plug gets flooded, could it still be gummed up?
A 2 cycle snow blower? Are you sure? So it has no oil sump, and you have to mix the oil with the gas?
2-cycles don't have compression releases. If you don't feel any resistance when you turn the engine over, something is very wrong. A healthy 2 cycle engine should have at least 110-20PSI of compression.
Assuming the problem is little/no compression.. If it's not a 2 cycle, then the problem is that you thought it was a 2 cycle.
If it is, it's either:
1) You used too little oil in the mix, causing lubrication break down and scored cylinder walls.
2) You used too much oil in the mix, causing carbon buildup and eventual cylinder wall scoring when pieces break free.
3) The engine has reached the end of it's useful life and needs a top end rebuild, typically ~300-500 hours for residential grade 2 cycle engines.