I am not all that sure that tradition does win over modernization in "Everyday Use." The story is more about what people do with their traditional ways and artifacts, really, since the characters either maintain traditions by incorporating them into their everyday lives or they relegate their ways and objects to the past, something to be looked at but no longer of everyday use. To the degree that this is what is meant by "modernization," certainly, Alice Walker's message is that tradition is preferable.
What we see in the story is that Mama and Maggie incorporate traditional ways and artifacts into their everyday lives, for example, actually using a quilt, rather than hanging it on the wall, which is what Dee wants to do. Is a butter churn an object d'art or is it something that should be used to churn butter? These are the dichotomies set up in this story. Traditional use of traditional objects triumphs in the story because Walker makes clear to the reader which characters are living good lives, content with their traditions and true to themselves, Mama and Maggie, as opposed to Dee and her new boyfriend, who seem to be remaking themselves and, while not abandoning the past, distancing themselves from it as much as possible by hanging it, figuratively and literally, on the wall. Mama and Dee are far more sympathetic characters, shown to be authentic and true to themselves, while Maggie and her boyfriend are shown to be discontent and insincere.