Which are the most authentic Mustakhrajat on the Two Sahihs, what are the benefits of classifying them, what is the criterion for accepting from them, and is it permissible to accept from them and then say, "narrated by Al-Bukhari or Muslim"?

The scholars did not specify the soundest of the mustakhrajat (extracted works), but rather some of them specified the soundest ahadith within them. The purpose of the mustakhrajat is the elevation of the isnad (chain of narration), not necessarily authenticity. Al-Nawawi mentioned that their benefits include the elevation of the isnad and the addition of authentic narrations, but Al-Hafiz objected to applying this universally to every isnad. Al-Suyuti added other benefits such as strengthening the hadith through the multiplicity of paths, clarifying the hearing of mudallisun (those who practice tadlis) or those with mixed narrations, identifying ambiguous and unnamed narrators, and ensuring safety from the defects found in the Two Sahihs (Bukhari and Muslim). Taking from the mustakhrajat is governed by the authenticity of the isnad between the extractor and the shaykh whose isnad converged with the isnad of the author of the Sahih. It is not permissible to attribute the hadith directly to Al-Bukhari or Muslim when quoting from the mustakhrajat unless the extractor explicitly states so, or if it has been compared with the original source.