The ignition, spark plug (properly gapped), and 2-cycle gas mix are all new. Why won’t my snow blower start?
The reason I say that the ignition is good is because upon turning it over (with the spark plug uninstalled but hooked up to the spark wire) it is showing a spark.
What next, check the carburetor? Could the carburetor be dirty? How do you get it off, that big nut (bolt) in the center of it?
The snow blower is a MTD® Yard Machines™ 4.5hp/21 2-cycle, Model #317E150. I believe the engine is a Tecumseh®, Model #HSK850. The snow blower is approx. 10 years old.
Thank you.

prime the carb, and insure the choke is closed. if needed spray some starting fluid into the air intake. if this don’t work, probably the carb is all gummed up
Try a short burst of starting fluid directly into the carb’s throat, with the choke off. Try to start it immediately. If it has spark, it will fire. once it fires put the choke on, until it begins to warm up. You may have to do this a few times until the bad gas is cleared out and the new gas is in the carb. If it still wont stay running, you may have to clean the carb. Starting fluid is hard on the cylinder walls and piston (clears out the oil) so don’t use it anymore than you have to.
Did you pull the choke on?
If everything else fails, take the carb apart and free the float and needle valve. If there was gas left in the unit over the summer and the gas evaporated, the needle valve may be varnished in place and not allowing the gas into the throat of the carb. To prevent this in the future, shut off the gas line first then run the unit dry, this will open the needle valve and let the float work correctly.
If the plug is old try a new one, sometimes old plugs will not spark when installed and under compression. If you can prime the carb, it should at least fire, meaning it’s not the plug. Then go to the carb repair.
Try pulling the plug and dribble a teaspoon of gas into the port, reinstall the plug and try starting…may take a few attempts to get it to run. Carb bowl may have some grit or varnish in it…try draining the bowl if it has a spring-loaded drain plug on the bottom or remove the bowl and spray it and carb passages out with carb cleaner. Check choke operation and primer bulb also. Always use fresh gas and put fuel stabilizer in it (Stabil) when not using.
check the compression and gaskets