Could be water in bowl. Do simple test first. Take carb bowl off dump it and wipe it out. Put back on and try again. Just a couple drop of water can cause this to happen.
You have never said if u tried jumping it with cables to full out bad battery. Could be electrical or could be mechanical. U need to first rule out which one it is.
Typically snow blowers don’t have an air filter, since there is no dust stirring up in the snow. Sounds like a simple carburetor blockage. You can get one on Amazon for $15.