If you look at the front edge of the lower shielding, there is a little squarish flap of metal just to the right of the sleep light (assuming the shielding is right-side-up). This should be vertical, and should rest against the front of the frame when installed. However, it often gets flattened against the rest of the shielding, in which case it presses against the underside of the board when installed, and often causes a short and prevents the machine from powering on. That's the first thing I look for when I have a machine behaving as yours is.
Beyond that, the hairdryer/heat gun/etc. approach to the video issue is about as precise as driving in a thumbtack with a sledge hammer, and the issue almost always comes back in one form or another. It might last 5 minutes or a month, but believe me, unless you've had the chip professionally re-applied by someone with a very expensive rework machine, you're going to be seeing related issues again soon.
But hopefully it's just the shielding, in which case you'll have a usable machine until video-related problems start again.
Good luck!
John