It doesn't really make sense to me for them to remain on the list after protesting their inclusion on the list.
Because whoever anonymously published the list never made a second edition
From J.K. Rowling's "Pure-Blood" writing:
In the early 1930s, a 'Pure-Blood Directory' was published anonymously in Britain, which listed the twenty-eight truly pure-blood families, as judged by the unknown authority who had written the book, with 'the aim of helping such families maintain the purity of their bloodlines'. (...)
A minority of these families publicly deplored their inclusion on the list, declaring that their ancestors certainly included Muggles, a fact of which they were not ashamed. Most vocally indignant was the numerous Weasley family, which, in spite of its connections with almost every old wizarding family in Britain, was proud of its ancestral ties to many interesting Muggles. Their protests earned these families the opprobrium of advocates of the pure-blood doctrine, and the epithet 'blood traitor'. Meanwhile, a larger number of families were protesting that they were not on the pure-blood list.
The Weasleys protested their inclusion after the list was already published. Seeing there was no subsequent versions of the list made, their name remained on it.