iPhone keeps rebooting apple symbol, and shuts off

I bought an iphone 5 on Craigslist and it worked fine until the battery started acting up, so there was this battery recall so apple fixed my iPhone a few weeks ago with a new battery. Then like a week ago, my I woke up the next day and then my phone started flashing the apple symbol, and I was SO frustrated cuz it won't turn on. It just keeps flashing the apple symbol. So I took it to the apple store and they said my Motherboard/logic board failed due to normal phone usage and they said they'd have to replace my iPhone for $290, and its frustrated me because I  just wasted my money. Now, my question would be could you fix/ replace the motherboard? like I looked on the Internet and they said it could be because of the battery replacement apple did, but Im not completely sure. Don't even try recovery mode, because I tried doing that and resetting my iPhone but it was unable to do so, signifyin a hardware problem.