A large proportion of adults will tell you that they "can't sing." Many of them believe that they are genetically incapable of singing, that no tune could ever come out of their mouths. I don't think this is true of most adults, any more than it is true of most children. The problem is not that they can't sing; they are just out of practice. People who practice something usually start out being terrible, but practice is about getting better with time.
Have you ever listened to a recording of shape note singing? I don't mean a choral arrangement from one of the folk songbooks like The Sacred Harp; I mean a raw, in-the-field recording like those made by Alan Lomax in the 1950s. It doesn't sound like typical recorded music. It's not pretty or polished; it's not something even I, a lover of shape note music, would probably sit and listen to. That's not what it is for, though. The Sacred Harp is a hymn singing tradition; the recreation of a community was to get together and sing. The shape notes were written to help people who couldn't read music learn the tune. There was (and is) no audition, to separate the "musical" members of the community from the "tone deaf." Everyone got together to worship, and the way they worshipped was to sing together. Over and over. Not for outside listeners and not even primarily for each other, but for the Lord.
Anecdotally, everyone has a story about the choir soloist or random person who sings out confidently yet isn't carrying a tune. Besides the fact that anecdotes are not data, though, I don't contend that someone who can't carry a tune should simply sing louder! Singing by oneself a capella is actually one of the more challenging things one can do; lots of people who find it hard to do that can sing better, if they find a place where they are welcome to practice doing so, and to learn to listen to other people singing with them.
In my family, we've been lucky in that we've long enjoyed sitting around and singing together, purely for the joy of doing so. Some of my relatives, to be sure, are particularly gifted singers, but as far as I can recall that is not why we sang in the first place. We sang regularly, partly because we were brought up in church where singing together was a central part of congregational participation. But above all we sang regularly.
The prevalence of recorded music everywhere, not to mention "idol" shows, has skewed contemporary perceptions of what singing should be. People don't think about the fact that a recording is almost never one live take--that even professional singers try many times to get a recording exactly right, and then it's further tinkered with in a studio. They hear singing "perfection," and imagine that's what should come out when an ordinary person sings. When it doesn't, naturally, they clam up and decide they are tone deaf or can't sing.
Even most community singing groups are self-selecting, in that people who already like to sing tend to join them, and people who've convinced themselves (or worse, been convinced by others) that they aren't musical do not. In the case of a choir with auditions, whose principal purpose is performing, it makes sense to start with a more experienced level. But as someone who grew up in a tradition where everyone sang--where the principal purpose was not to sound good for other people, but to worship God--I believe that singing, like exercise, is good for everyone.
There are hard data (not anecdotes) showing that singing has physical and psychological benefits similar to yoga, but that's not why I'm comparing it to exercise. Think about this. Many people who could greatly reduce their health risks by getting more fit are reluctant to exercise, because they don't think they look good exercising and feel self-conscious in front of more fit people. Or, they aren't gifted at sports or have been made to feel bad (sometimes unwittingly) by more athletic people in the past, so they are reluctant to try. Not for a moment am I suggesting that these people should immediately set out on a run; they might injure themselves and then be unable to exercise at all. But absolutely, I think everyone should get the benefits of whatever exercise they can do--in many cases, just going out for a walk. Does anyone seriously contend they should not, or that if they do, their fitness and coordination cannot improve with time?
Singing is like learning a language. Some people pick languages up more quickly, but almost everyone learns to speak one: by listening and being willing to make mistakes.
We need to rethink what singing is for. For some people, it is about being an entertainer, but for most of us, it's an activity to enjoy and to value--physically, mentally, socially, spiritually. Singing folk music, or Christmas carols, is different from professional performance; it is about enjoying the music and participating in it oneself. Of course, as we sing more regularly and learn to listen to others, we will more often get the right notes, in the right time. But the only way to do this is to open our mouth and let our voice out.