It's possible linux will fail but give it a try using Rufus. You don't have too much to lose at this point. Replacing a motherboard on a laptop is possible but takes some good computer skills and patience plus a good service manual for your model helps. As Captain_Chicken suggested you may be better off purchasing a new computer.
I primarily have worked on PC desktops, and have had no issues building up my computer that way. I handle all the maintenance on our desktop, and have swapped out every part under the sun on it thus far. I've had to replace motherboards in the past, and have done upgrades to most of the other internal parts at this point. Everything is good on that end, so I'm confident in my ability there- but it's also not a Toshiba. lol
I just don't have as much experience with laptops, because I frankly have not had much need for one up until this point. Every laptop I have ever had to fix with the exception of one, was a toshiba satellite. They seem to be poorly made and frequently have problems like this. Had I not been given this laptop for free, I would have never bought one of these. I would literally go to Walmart and buy an Acer, before I'd buy a Toshiba. lol.
I'm really not trying to invest a ton of money in a laptop, but would be willing to buy the part to fix it if needed at some point. I've tried everything I know how to do thus far, and this problem has me stumped. Everything I've seen so far has led me to believe that this is a hardware issue, but I figured I'd check in here to ask, since I don't work with toshiba very often. I just found it odd that it crapped out like that during an OS reinstall, after working fine. My only thing was it had been sitting in my office storage untouched for about a year. I'm wondering if just sitting idly contributed. lololol. Seems like you breathe the wrong way on a toshiba and they quit working.
I will give Rufus a try this evening though, for sure.