I'm trying to determine if this is a fuel problem or spark problem and the symptoms can indicate either one. Many times when an engine is running out of gas (lean) it will surge and sometimes pop (backfire) due to running lean. I'd really like to see an inline spark checker hooked up while the problem is happening.
Have you replaced the fuel lines? I have seen where fuel lines soften and squeeze together shutting off fuel flow to the carburetor and this can cause an engine to surge then die. I have also seen an obstruction in the fuel line cause similar symptoms. The list can go on for a long time and I don't want to recommend a part change without proof that it failed. A 14 year old mower could have something bouncing off the bottom of the fuel tank such as a leaf that will intermittently plug the fuel tank outlet. There are so many possibilities in a case like this and until you can pinpoint something it is just speculation.
Until we determine what factor is missing when the engine acts up it is going to be a shot in the dark checking all the systems on the mower for the actual problem.
The fuel pump has an impulse line that connects to the engine. Find the line that is not one of the two fuel lines and check it carefully for leakage of any kind. This line provides the fuel pump with alternating vacuum/pressure to operate the fuel pump. If there is a weak spot in the line it could collapse which would cause the pump to not pump fuel. Again, I'm offering possible causes but you will have to check all these items to verify if they pass or fail.
I'll keep thinking on this one and if anything else comes to mind I'll make sure to post it here.