Fixing my Dell Inspiron

So, about the laptop in the title
A buddy called me because his machine keeps shutting down. Some repair guy told him the motherboard is shot, $133 repairs and all that nonsense...

I take a look, and man, the CPU and board are cooking (sitting at 70-90 degrees Celsius).
(My bad, I should've checked the fan first.)
I pop the back panel off, and the heatsink looks spotless, but the thermal paste? It’s basically evaporated. There's almost nothing left...

I clean everything out, slap on some fresh paste, and put it back together. The fan won't budge (it only has one)...

And that's where the headache starts. (I didn't have my multimeter on me at the time.)
I tried measuring the pins with an old meter I had lying around—there's some variance there. You've got the black and red (+), plus a yellow one that I can't quite figure out.
The fan is dead in the water... the resistance is bleeding over to the black and red wires.

He takes it to a shop, they find a used working fan, plug it in, and still... absolutely nothing happens.

How do I test those pins? I'm trying to cool the laptop down just enough to see if I can get macOS running smoothly again, just to rule out some weird software setting being messed up somewhere...

If the outlet on the motherboard actually died, what's the easiest way to wire up an external fan to keep things cool, since everything shares a single heatsink?

The laptop is strictly for web browsing, no gaming or anything heavy, so it doesn't need massive cooling. The real issue is when he's scrolling through Facebook—it overheats on white backgrounds, which is a huge pain.😁

So, how do I cool this thing down, and what's the best way to test the motherboard pins...
The fan runs on 5V.

I'm heading over to his place today to go through this properly with a real multimeter... we'll see what the verdict is.

If you can't find the transistors, swapping them is a nightmare..
I once rigged a 5V power supply by connecting a fan through an 80-Ohm resistor,
which gets it spinning at about 3 volts. It might be too slow—you never really know for sure—so just grab a 40-ohm one instead. Polarity doesn't even matter here.
Normally, I wouldn't bother discussing this unless you were an electrical engineer, but this isn't actually that difficult.
There is no cheaper fix, and it works.
The fan needs to cool things down, but it doesn't need to run at a full 5 volts.
It would run quietly and still provide enough cooling. I'd bet it stays under 140 degrees.
Maybe the heatsink is clean now, but I suspect it was filthy, perhaps because it was used on a bed, forcing the fan to run at max capacity... who knows.
Too many people understand nothing about tech, yet they buy everything without ever reading John Doe..
I knew a guy who bought a car from my brother years ago... and I watched him run it into the ground.
He never changed the oil, never topped it off, and never checked the coolant... he just went to the gas station for fuel and that was it. He had nothing left.
It's the same with how people neglect cooling on PCs (especially notebooks), fail to replace the CMOS battery, or never clean the power supply inside a desktop case.
I couldn't read or reply until now; I hope you actually see this..