Read thru all posts. I was leaning toward fuel but I think you exhausted all fuel issues (minus replacing all fuel lines). I don't think they are collapsing or sucking air or tractor would run crummy bit not just shut down. A '99 would have a Low Oil Pressure shutdown. Wondering if that switch is seeing a false 'low pressure' when engine oil heats up and viscosity changes.
One would think that porous fuel lines would not cause the symptom the OP is experiencing, but it can and did with my situation.
Having a "Mushy" fuel line, regardless of where it is on the tractor, tells me these lines should be replaced before doing anything else.
I am not a New Holland/Ford 1920 owner so I may stand corrected, but fuel lines are fuel lines, defective from the factory or just old.
Replacing the fuel lines is good solid advice and can be done relatively cheaply and easily unless you have a Mahindra 5035, which required removing the rear wheel for me to get my somewhat large hands in there to do the swap.
Changing the fuel lines before doing any of the other recommended procedures such as draining the tank and inspecting same with a borescope saved me a lot of needless troubleshooting on a fairly new tractor, as noted in my post back in April.
If I read your response correctly, the OP has not replaced the fuel lines yet. Hopefully, this will be an easy repair that will resolve the problem. I may be all wet on this but I thought it was share-worthy as my tractor was behaving the exact same way... no sputter or rough running, just stopped. It would restart after sitting for 20 minutes, or I could force fuel into the system manually using the primer pump on top of the fuel filter housing.
If this doesn't resolve the problem, it was certainly time well spent considering the age of the existing lines.
Ng