Apple say this to just about every customer when a MacBook wont turn on. They dont know why it wont turn on but they have just checked it isnt the keyboard, trackpad or battery and so logically it must be the logic board.
Yes, in this case, there will be a fault on the logic board. Apple's solution is always to replace the logic board with another and its the same decision that the vast majority of computer shops would make [but this is only because they do not have logic board repair skills].
It has been suggested that Apple's very high repair fees are designed to encourage purchase of a new MacBook. That might explain why they wont quote for items over the phone insisting you come into the store - you can choose a brand new MacBook there when you hear their high price for a repair or a new screen or keyboard.
In my opinion Apple are just not geared up for repair and dont want to be - their mission is to sell new computers. They only offer replacing the logic board - and not with a new one! For this they charge nearly 80 to 90% of the cost of a new machine and will take many weeks for it to be returned - who would prefer that than purchase of a new machine for a couple of hundred dollars more.
Its psychological - they are pandering to your desire for retail therapy!
Fight it! Dont buy a new computer when you could get your present one repaired far cheaper than Apples quote. When I say far cheaper I mean for a quarter to half the cost quoted by Apple.
My recommendation is find an independent repair shop that advertises Macbook logic board repair then ring them and ask them about it to be completely sure - you can save yourself a lot of money and have your MacBook purring away in a couple of days rather than months.