How can the Sayyids be descendants of the Prophet Muhammad (PBUH), knowing that he had no sons and that lineage is traced through the son, not the daughter?

All of the Prophet's ﷺ grandchildren and descendants currently alive are from the lineage of his daughter Fatimah, may Allah be pleased with her. They are attributed to him because he is the master of humanity, and this is one of his ﷺ unique characteristics. Scholars have inferred this from the fact that the Prophet ﷺ called Al-Hasan his son, even though he was his daughter's son, as he said: "Fatimah is a part of me."

This specificity in attribution to the Prophet ﷺ is confined to the children of his daughters, meaning the children of Al-Hasan and Al-Husayn, and does not include the children of his daughters' daughters. A child follows his father in lineage except in this special case concerning Fatimah's children. Therefore, the son of a Sharifah (female descendant of the Prophet) is not attributed to the Sharifah unless his father is also a Sharif.