I think Cinderella speaks to the hopeful side of humanity on a universal level. The protagonist is consistently put upon by her family (and who doesn’t feel that way at least some of the time?), she goes behind her guardian’s back (who doesn’t at least want that sometimes?), with the help of an ethereal being who turns ordinary things into priceless things (could be analogous, and definitely desirable), pretends to be someone she isn’t, captures the love of a prince (what is it with princes?), and eventually goes from being a good hearted peasant to a good hearted queen.
The elements in Cinderella have been used in movies, books and songs for a very long time. According to the Cinderella page on wikipedia, it is one of the most remade stories for movies of all time, with a cinderella-esque movie coming out about once every year since 1899.
The rags to riches idea appeals to us because it throws the idea into an area of possibility, and the idea that we can achieve our dreams by doing no more than having a good heart and crying out when we hurt, is beautiful in its simplicity and marvelous in its poetic justice.
The prince at the end is just the prize. I don’t think it’s so much that we’re teaching our daughters that they should wait around until some singing nobleman on a white steed carries them away from turmoil as it is us teaching our children that when you do the right thing, are compassionate and nice, you are then the ‘good guy’ and in the end you’ll achieve your goals and dreams.
The persistence of Cinderella is the idea that we are not fated to failure by our circumstance, but destined for happiness through our integrity.
And I think that appeals to everyone, not just children.