Overheating. Is it the fan or the main board?

My Air wouldn't start up, or would shut down within seconds of lighting the screen. After letting it sit, it started up to the point of login. Then it shut down again after a few minutes.

It's charged. I've done the resets, and all the goodies suggested on Apple's site, but still it doesn't consistently start up.

I am leaning toward it being a temperature problem at this point because I started up and did the Apple Hardware Test and ran through the simple test a couple times while holding the laptop in front of the air conditioner. All tests passed and it didn't shut down.

I've opened it up, cleaned it out (it wasn't bad), and even replaced the thermal compound, but all to no avail. It's still doing the same thing.

One thing I've noticed is that the fan never turns on. The bottom is not getting excessively hot (I've felt much hotter), and I don't think it'd overheat within a minute of starting up, but perhaps.

So, I'm wondering if there's a way to identify if the fan should be turning on and just isn't (a bad fan) or if there's a problem with it reading the temperature, or if I just got unlucky and need to have the main board replaced.

Thanks!