I tried to teach a friend to sing in tune once, and we did make some progress.
I started by simply playing two notes, and getting her to listen for same/different. Then I played her scales and glissandos and showed her the concept of lower to higher, and got her to play them. Then I played various notes, and had her listen to whether they were generally 'low notes' or 'high notes'. Then tried to get her to say whether a second note was lower or higher than the first, both with very wide intervals and then narrower.
Then she did some glissando type things with her voice, actually trying to make really low notes and up to high notes, up and down, to get the feeling of how to change pitch in her voice. I didn't have one at the time, but now I have a tuner app on my iPhone that shows the pitch you are singing (well, it wobbles around a lot!) but you can kind of get the feeling if you are making your voice go up or down, which might help. She didn't really have any idea how to change what her voice was doing at first.
Then I'd play a note or sing one, and just try to get her to match it - and I could point up or down to encourage her to change to match mine, listening to what it would sound like when it was together, and what it would sound like when it wasn't. (I could also play a note, and sing a different one, to show her what it sounded like when it wasn't the same note).
She was only visiting for a weekend, and it was just an experiment to see if she could improve, as she'd always loved the idea of singing, but was really quite tuneless. And she did improve. By the end of it, she was matching my pitch about half the time, after starting out wildly different. I think with more practice, she could have done it, and then started to learn to reproduce intervals. We were not working on fine tuning, as in whether she was slightly flat or sharp.
I'm sure children could be taught to do something similar, especially with someone who knew what they were doing in terms of both teaching music and teaching singing, like how to actually create the high and low pitches (I am not particularly good at singing or being in tune myself, but I have a general idea and can play instruments).
children's choirs often seem way too high when I hear them, and I think no wonder half the time they sound flat! Softer, lower, and lots of emphasis on listening and blending would help.