I just got this Mac from my grandma because she said it wasn't working. It boots and lets me go into recovery mode but crashes in a similar way whenever a loading bar starts. I've tried booting every different way and cleared nvram. I didn't see any clearly visible damage on the logic board, and I'm willing to try any ideas as long as I can keep the price of fixing this under $100.
Talk to Apple.
They can do a remote diagnostic to figure out where the issue is. Also try booting in safe mode. Hold down the left shift key while booting.
It does sound like a logic board issue though.