The red tube just extends the bowl vent. Most likely the carb bowl ridge is corroded and not sealing. A new bowl w/ seal is likely the fix. new emulsion tube seals may be in order as well.

the bowl and seal looked pretty good when i was cleaning the carb. the seal didnt have any cracks and was still pretty 'rubbery'. so the red tube leads straight into the bowl? any idea how blowing air into it makes it work? is the air unblocking something temporarily, which then gets blocked again when i'm cutting the grass?

Blowing air is a old saying tickling the carburetor. You are enriching the gas by raising the float level and kinda flooding.

I assume this carburetor has a primer bulb?

Anyway, I would check the float level setting. The model number of the carburetor or engine would help me.

And if you look at my other posts you will see that you can get a China Knock off cheaper then replacement rebuild parts.

Also if it is a fixed jet you may have the bottom hole partially blocked.

Anyway, most people forget to set the float properly.

here is another way of describing what the others have stated.

pushing the primer bulb pumps air into the float bowl. A small amount of fuel is displaced through the red tube into the venture area. This provides fuel for starting. When the engine fires, fuel continues to be drawn into the airstream in the venture. The main jet meters the correct amount of fuel and the engine is happy.

In your situation, the primer is not delivering the correct amount of fuel. A puff of air provided airflow through the venture and drew fuel from the floatbowl. When emission requirements were developed, Tecumseh designed a primer that was right on the edge. If the fuel level in the float bowl was not correct, or if the bowl gasket had a tiny leak, the primer will not deliver enough fuel. Additionally, the primer bulb may also have dried up and collapsed slightly. O rings on the venture tube may have dried up and cracked. Ethanol fuel contributes to deterioration of the rubber parts.

Some Tecumsehs have a very small orifice in the fuel bowl bolt. Take that brass bolt out and clean it. Look for a tiny bleed near the top of the threads that looks like a speck of dirt. Take a small wire tie, pull the paper off, straighten it and use it as a rod to clean the hole. This time it should run.