Let's put thing in perspective here.

There's not much value in 2008 Mac Pro. I sold mine for $450 USD with 16G Ram, SSD, upgraded GPU.  If i had have a bad logic board, ordering a new one would cost me probably 3-4 times more it values. I would rather not fixing it if i can't find a used part.

I would call in Apple center and find out how they charge on diagnosing and repair cost. Then I would bring in local computer stores who has spared parts and can pin point the issues. If they manage to find out the problem and bad hardware, then you will decide what's more cost effective. Save the money, buy a faster new Mac or bite the bullet.

But don't give up yet. I went though a lot when the GPU went out, i was ready to throw in the towel on logicboard or riser card and turn out it was the GPU. I was lucky to find a guy would let me test on his spare video card before buying from him.