I totally agree that tutting at someone is rude. However, knowingly annoying others in a public setting and doing noting to mitigate that is rude. Singing in a restaurant is, at best, right at the edge of rude. Anyone, small child, adult or otherwise with a high pitched voice singing in a restaurant is definitely rude. To allow your child with a high pitched voice to sing in a restaurant and then get upset that people judge you is absurd. Polite behavior would be to teach your child that there is a time and place for singing, that doesn't include in a restaurant.
I'm not sure what you mean by, "I know it's hard, but your responsibility is to your child". I hope you mean that your responsibility is to teach your child the right way to act and that having a tantrum or singing in a restaurant is not acceptable.