To be fair, the question was about logic, not emotion.
Yesssss. Mate I understand that, but if my post is clear and you read all of it, I'm NOT saying that emotional investment is a good thing in making your decision, rather, I'm suggesting that many players, such as 5ynthetical, who you quoted above me are now FIRMLY locked into their positions on the ending because they 'like' their ending.
I'm suggesting that by even CONSIDERING trusting ANYTHING this star child says to you in the first place when the reapers are backed into a corner is made invalid by the fact that he is "the collective consciousness of the reapers".
I'm suggesting that others who pick the 'non-destroy' endings are heavily influenced by two emotional processes:
Firstly: Their attachment to the Geth and EDI who they don't want to kill off with the destroy ending. This however is assuming the reaper child isn't lying to you in the hope to dissuade you.
Secondly: Their desire to not feel like they have 'Lost the Game'. Because IF Andromeda ever confirms a canon destroy ending, where control and synthesis are made officially wrong, they are going to FEEL really sad and angry, because that would mean they officially lost the game.
As star child freely admits, he is the collective consciousness of the reapers, so he IS the reapers, but in the form of an innocent looking young boy to make the player confused. Now The question of whether this thing is EVIL or not. I don't care. As the reapers would say 'Irrelevant'. Because it's not about good or evil, it's about TRUST. You CAN NOT trust your enemies information. Star child, evil/not evil debate aside, is your enemy and you're going to TRUST the things he says?
Are you telling me that after all the times the reapers have tried to kill you or desperately pursue and destroy your ship in those resourse scanning mini games, that suddenly, when you have the chance to destroy them, they're letting you make three choices out of benevolence? No. The choices are presented by the reapers in desperation to try and encourage you not to destroy them. The whole ''Ohhh you'll destroy ALL synthetics" thing? Isn't that convenient?
In behaviour sciences, one of the ways in which you try to stop an angry child from having a tantrum and demanding, say a lolly is through providing them with the choice they want minus something else (lolly now= no ice-cream after dinner, that is: Destroy= ALL synthetics destroyed), then an alternative (eat a piece of fruit, then we'll go play in the park this afternoon AND you get ice=cream, that is: Synthesis, make a lesser sacrifice, stop the reapers, save synthetics). I couldn't bother thinking of a metaphor for control, but the idea is to try and dissuade the child by presenting more attractive options and painting the original option as less appealing.
Like I said, my main point is that if you want to use PURE logic about this, there is NO means of validating ANYTHING the star child says. He even tells you that he is "the reapers" (see my previous post for the link, 2 mins 30 secs into the video).
You can still save yourself

! Save your Shepard! It's not too late! Don't let the Reapers win! Come back to Destroy before Andromeda! We destroy them or they destroy us.