A vacuum diaphram fuel pump with a tank lower than carb level...bad idea !
Just about every JD tractor mower made in the last 20 years has this system and they work just fine.
Add to that about 1/3 of the output from AYP ( Husky, Craftsaman, McCulloch, Roper, Rover etc etc )
They all work just fine so there is nothing wrong with the design.
There is a problem with your mower and to get to the bottom of it we need to know EXACTLY WHAT IT SOUNDS LIKE WHEN IT STOPS .
Otherwise we will have you chasing shadows for no good purpose.
So to start we need to work out if it is a fuel or electrical problem.
It could be some thing as simple as water in the tank or some thing as difficult to find as a loose wire making contact when the mower is on an incline.
Now it should have a 22 Hp 44N677 Briggs engine in it which according to the IPL I have does not come with an oil level switch but to confirm this we need the rest of the engine number .
Now the rest is up to you.
You have the mower in front of you.
None of us are there drinking your beer and testing the mower so unless you answer the questions asked then we can not really be of any meaningful assistance.