Why? If these kids don't want to feel the sadness, maybe pick the right song that highlights their abilities at the beginning? (It's already "forgiving" enough that they get to sing until the end, versus a resume where your app will be tossed if you can't impress in the first 2-3 sentences). And I say kids, because adults should be able to handle rejection. In this industry! Heck, parents can encourage and praise their kids singing all they way, but they should also prepare them for the real world - "you are not going to get a yes every single time."
I have no idea how the singers pick their songs. Do they select from a list of "approved" songs, or can they sing whatever they want? To a certain extent, of course; no one is gonna be allowed to sing a song filled with inappropriate words. For the luckier ones who got a chair to turn despite their shitty song choice or shitty singing, maybe the judges didn't hear anything they cared for until a glory note toward the end, or an amazing run after the chorus, or a pity button press.
I know for me, that one girl who sang Jewel's Foolish Games - 5 notes in, I was saying to myself "wrong song, girl". She may be an amazing singer, but I'd never know because she picked the wrong song for her voice; plus there's nothing in her presentation+personality that made me want to go youtube her name. In fact, I found her bitch-face right after singing to be rather off putting.
And then there's shenanigans like Matthew Schuler, a 4 chair after like 8 seconds of singing.