Did you read my earlier post???
Any way, remove the fuel tank, pour the fuel into a clean container, then shake the tank, in a quiet area, with no radio or tv to distract you. Listen for anything rattling in the tank! If you hear a rattle, turn the tank upside down and shake it to let anything fall out the fill hole. Then stick something in thru the fuel outlet, to see if it is plugged. install a new cap!
Replace the rubber fuel line. Remove the carburetor, take it apart, check the float to see if it has been crushed by water freezing in the carb bowl.If crushed, replace it.
Then, soak the carb in carb cleaner, or laquer thinner. Blow out all teeny holes in the carb. reassemble carb. try to start it.
If it won't start, here's what to do: Either take it to a repair shop, or, if you can't afford that, set it out front with a For Sale sign on it, and get rid of it! Buy a new one, or a large snow shovel!

rebuild the carb, install fresh gas

I hope this isn't too late.
I'm not sure where you got the idea that there should be a "primer button hose", perhaps from other engines you use. Are you certain you had one previously? You do mention that it was worn through due to dry rot, so presumably, there was something physically there that you saw and removed (the remains of). Are you certain it was worn through, or is it possible there was supposed to be a hole in it? Again, was this "hose" original to the machine, or did you buy it used or did someone else tinker with it?
Here's the thing: there is no such part on the old MTD Snow Blower I have (Tecumseh engine 155006), nor in the engine schematic or parts manual. Note also (on mine) the instructions indicate to "press AND HOLD" the primer button while "pulling the starter rope slowly" and to do so only under very cold starting conditions. I've never had to do this. I believe the purpose of this button on this particular engine is to restrict the air flow when depressed (it is attached to a small steel plate which, when depressed, completely seals the air inlet regulated by the choke) unlike other (such as on my lawn mower) primers which utilize a flexible bulb to force a volume of air into the bowl to force fuel from the bowl into a fuel/air mixture passageway. Notably, the "primer opening" on my engine is normally wide open. I suspect your "rubber hose" has sealed it off permanently on yours, and that this is contrary to the proper operation. This might explain why it worked when you had the hole in the fuel hose, which then allowed air to enter. On the other hand, you may have a different model and I may be completely off-base here. Perhaps someone who has a better understanding of older model engines can enlighten us.
Best regards.

Two different types of primers. The type zmdgsno is talking about, indeed has a hose that is attached to a bulb which draws gas into the bowl. The type you are speaking is not really a primer, but a plate that occludes the air into the carburetor.

Since this was posted over a year ago, I hope zmdgsno found a solution to his problem. My guess is that his original problem was a stuck float. Pull the bowl off and move the float back and fourth.

Just a thought here. Could the breather hole in the gas cap be plugged? This would cause the problem.