I understand the dilemma you feel you are in; my kids are 13 and 17 now and I have definitely had to break the news a few times.
However, I think that singing is a whole different animal than, say, soccer…music is an integral part of who we are. Whether or not we can match pitch has little to do with developing a VERY healthy and important relationship with music. There is strong evidence that kids who play a musical instrument (and read music) develop enhanced verbal memory and literacy skills. There are links between music and improved math skill. Music can decrease the likelihood of clinical depression.
And it is the ultimate bringer of joy. Clearly, music brings your daughter joy. Let her sing. Badly. She will eventually either get better (this is possible) or she will figure it out on her own. I promise. My son certainly did. And just fyi all that did was keep him from literally signing up for talent shows as a singer. He still sings in the shower, and along with the radio, and with his headphones on. And he WHISTLES INCESSANTLY. And totally off-key.
I am freaking Whistler’s Mother. No shit.
Telling her not to sing is bound to take away at least some of the joy music provides her, and for her whole life, that piece of information will have YOUR face all over it. Nobody wants that.
Let the REST of the world be the bad guy. You be her soft place to land.
“A bird doesn’t sing because it has an answer. It sings because it has a song.”--Maya Angelou