Well, I don't know about the belief the teeth wear and tear at twenty, but teeth effect singing from a different aspect. However, teeth cannot affect your singing so much that you are a bad singer.
Anatomy in general can affect the sound of one's voice. Some people (please do not laugh) who have big overbites or buckteeth often have issues with the shape of the vowels, and even projection of sound. They try to overcompensate often over the overbite with their mouth. Underbite can cause the same issue. I don't think braces do that much to the sound. You might have pronunciation issues in the beginning, but once you get used to functioning with the braces, it isn't really a problem. Most people with or without braces have to work with pronunciation anyway. That is normal for all singers.
Other than that, the soft palate, the tongue, all those things inside of the mouth other than the teeth attribute more to singing. Teeth doesn't do that much unless there is something dramatically different about the teeth. And nowadays with the invention of braces, most of us will not have that issue anyway.