Maybe, maybe not - cost me $250 or so to have motherboard replaced, but it was a $3800 machine with a close to $4400 current replacement cost, so for me it well worth the cost. I'll always risk $250 to Make $4400
OP's 10-yr old workstation likely worth ~$100 if working. (I bought an old T3500 on eBay a couple of years ago, likely same vintage as OP's, for $75 including ship.) Not worth replacing mobo.... not to mention the old technology.
My Core 2 Duo's got to getting upto 10years and will fail 1 day been over clocked since day 1 so amazed it's lasted this long. Got a 2nd Computer Game Spec, but can't be arsed to transfer everything, so 1 for work 1 for games. I'd say Chip gone as no beeps, once they over heat, new paste won't fix. Try pulling the ram and any other slot out and cleaning the contacts with a pencil eraser / rubber.