No idea how to test it aside from swapping things out or testing the electrical contacts one by one. To find one, ebay would be the best place to look based on your exact TV model. If you can replace it yourself, and the system board is actually the issue, it will be a lot cheaper than buying a new TV. Don't know how you feel about opening up the TV, most of the time you don't want to mess with that unless you are good with electricity as there are plenty of components in the TV that can give you a nasty shock if not put you in the hospital if you touch the wrong thing.
If you are looking for such a large replacement, even with paying someone to do the work it would be cheaper to fix the thing. If you get a 42"-50" or so TV, you probably should replace it. A new 80" size TV would be pretty pricy.