<human>: Give me the gist of the text in a nutshell

It's a Briggs & Stratton 6.5hp.  I got this mower brand new in 2003 and it has been a super reliable, one pull starter its whole life.  I have carefully maintained oil level and changed it out every season.  Every winter it stows in my garage under a tarp with a dry gas tank.

Well, spring has sprung.  So I take her out, put fresh gas in her and hit the primer bulb the magic 3 times.  One pull and she starts, runs fast, but then after about 3-4 seconds sputters and dies.  Huh.  I try again.  Same thing.  More primer bulb presses seem to yield a few more seconds, but the inevitable run-sputter-die pattern holds.

So I can't get this thing to run more than 5-10 seconds.

Hmmm.  Here's what I have tried:

* Changed the air filter
* Changed the spark plug
* Pulled the fuel line off the carburetor end and drained the tank into a gas can (confirming the fuel line isn't obstructed)
* Pulled the carburetor, disassembled it, it looked pretty clean but cleaned it anyway.  Using a drinking straw I am able to blow air through the jet.

Other observations:

* I don't smell gas, or see leaking gas
* There is no odd smoke or anything like that during or after the 3-4 seconds of run time
* The 3-4 seconds of run are very strong; no misses.  It just runs well for 3 seconds, then quits.

After putting everything back nothing is different.  Mower starts on one pull, runs a few seconds, sputters and then dies.

I'm just a caveman, but it really acts like it is only burning the gas that is pulled into the carburetor when I prime it, and then it can't draw more gas.  But this doesn't make sense because there is no obstruction in the fuel line or the carburetor.  So it's got to be something else right?

Small-engine repair Redditors, I need your help.  What else should I try?
<bot>: Normally reliable mower starts immediately, runs 3-4 seconds, then dies.  I have tried the basics.  Help.