Intermittent problem - engine starts missing and cutting out, almost dies.

Hoping someone can give me some idea what could be happening. I have a 2006 STT72A-35BV-SS Scag Turf Tiger. Have had for 13 years and has always run great. Recently lost two PTO switches until I realized the coil for the PTO Clutch was dying and finally went belly up. So I replaced the clutch and rebuild the mower spindles, replaced all belts, all idle pulleys. Not that these parts were bad but, 13 years old and had never been replaced. All was well and began mowing again thinking I should be trouble free.
Now the Briggs and Stratton Vangaurd 35HP model 613177 type 0114-E1 code 05122612 will run great for around and hour and then intermittently start cutting out and looses power and acts like it is not getting gas. I unload buy turning off mower deck and it starts running fine again but will keep repeating the problem every few minutes. If I let it sit and cool it will run great again for a good period of time (hour or so) and then start same problem intermittently. So I replaced the air filters, the spark plugs, the dual fuel pumps, the electric module, For kicks and grins the radiator fluid and the drive filter and fluid. Thinking it was the fuel pumps only to find again after about an hour of mowing it starts to pop and sputter intermittently again.
This is a water cooled unit and I watch the temperature gauge and it does not rise or lower from what has always been normal. I am at my limit I don't know what to do to solve this problem. The only manual I have is the operators manual that came with the mower when I bought it new. It has no engine information and is far from a shop/repair manual of which I cannot find on this mower. Can anyone point me in the right direction?

I would adjust the valves first.

Then hook the mower up to an alternate fuel tank when it starts to play up.
Floating debris gets pulled into the fuel outlets on the tanks when you are sucking large volumes of fuel.
This forms a plug which slows down the volume of fuel that can flow through the fuel lines so after a while the engine starves.
You can get a similar things happening at fuel taps on multi tanked mowers.

repair manual #275429

Intake .007 - .010
Exhaust .007 - .010

Fish said:

repair manual #275429

Click to expand...

Thank you I have ordered #275429 $40. I don't know what this covers but I will be very happy if it gets me past my problem.
Thanks again.

Fish said:

Intake .007 - .010
Exhaust .007 - .010

Click to expand...

Fish - FYI thanks for your advice My intake and exhaust valve ranged from .012 to .016. I now have them all set at .008. It will be a few days before I mow again to give it the test.
Thanks again, Marty

Since the motor runs fine for an hour, Bertmobile 1 answer makes the most sense to me. Rust in tank gathers to plug fuel exit. I have a tractor that does this. I would blow an low pressure air line back from the fuel filter into the tank to push the sediment away from the tank outlet.

Both have value
Valves that are too tight or too lose can & do cause problems as the engine heats up.
Valves being mechanical are easy to check & set.
Even if his problem is a plug the engine should run a whole lot better now with the correct valve lash.