Seen your short video and I understand the answer you got from Apple centre..since they don't do component level repairs they'd be changing the logic board. They might be right, some power sensing issue or the likes seem to to be the right answer here but only deeper lab testing can give a clear answer.
Try to start it with battery unplugged just not to let that untried, but probably it won't make a difference.
If you want to dedicate some extra time try also starting the Mac after removing one at a time the various connectors, lvds, battery sensor, trackpad and so on, to rule out possible shorts..one can never know for sure without trying.