Low oil light on start

I've just purchased my 2nd Discovery II. It has 111k miles. I found a project with known problems. The previous owner described oil leaks and coolant leaks which I've diagnosed as transfer case front seal and the throttle body heater. These issues don't worry me but I am confident the Disco has larger issues. First off, I have the oil light on for 2-4 seconds after start. I'm sure I need a new oil pump, nothing to worry about there. I am wondering though if I need new bearings(mains or otherwise). It had new head gaskets at 60k miles. That is the only engine work I'm aware of. The exhaust smells and looks clean. No coolant or motor oil seems to be leaking into the cylinders. The motor oil on the dipstick is clean, no signs of coolant in the sump. After the oil pump is replaced I'll fit an oil pressure gauge and run the 4 readings to assure the problem was just the oil pump and I have acceptable oil pressure. If it isn't and I need to start major tearing into the engine, what clues of major problems(other than the steam cleaned cylinder(s)) should I be looking for? I realize I'm mixing coolant loss problems and oil pressure problems, but would like advice on common or known weaknesses in addition to a pinhole in the block behind the sleeves. If I decide to go all the way to tearing down this engine, what should I absolutely address and upgrade? While money isn't a "no object" kind of thing, I don't mind spending $4k on this engine if I can be reasonably sure I can get 100k out of it. I'd rather build a bullet proof 4.6 than hassle with transplanting a diesel that will be under-powered and take a year to get right. I have access to wholesale or greatly reduced(and my labor is free) prices on tear down and assembly. I only need to send out parts for machine work. The body is immaculate, color exactly what I want, white on black interior. What are the best components to rebuild this engine with? Thanks for any input!

What year is your truck?
At your mileage I would replace the oil pump using a kit, replace the timing chain and gears, new 180 degree t/stat.
There should be no reason to worry about the rest of the engine as long as there is no upper engine clicking.
You might want to replace the o-ring for the oil pick up tube and if it were mine I would bull a bearing cap from the crank to see what wear you have.

2004 se7

The Discovery is 2004 SE7. The only noise is a whine from the front of the engine, belt noise or a pulley. I'll try to narrow it down once I get it back to TX. I'll definitely check the crank bearing while doing the oil pump, timing chain and the rest of your suggestions. Thanks for the input.