It doesn't sound good. You say it attempts to boot, but, the power LED starts to flash during the boot process? I wonder if anyone documented a "blink code", like automotive computers use.
I think I would hope it is an oxidized connection rather than a failed surface mount chip. Have you cleaned ALL of the mating connectors? You might examine all the chips with a magnifier and maybe sweep the legs with a dry toothbrush. Has the case been left open? I'm sorry this is not much help.
What's the story on the 2nd motherboard? Fortunately for you, I don't think A600s and their motherboards have yet begun to skyrocket in price.
There are many techs who praise the reliability of surface mount construction. Perhaps it is better when a computer is new, but, when you are trying to keep an old machine alive, sockets and thru-holes are a lot easier to work on.
Is your 600 expanded with HDD, acceleration, & other expensive add-ons?
