Why were fabricated and weak hadiths written or preserved?

Muslims documented weak and fabricated hadiths for several reasons, including:

For weak hadiths: 1. To preserve everything narrated from the Prophet, peace and blessings be upon him. 2. Weakness is relative. 3. It is permissible to act upon them in matters of virtuous deeds, under certain conditions. 4. It may be strengthened by a multiplicity of chains of narration.

For fabricated hadiths: To warn against them, so that no one narrates them and attributes them to the Prophet, peace and blessings be upon him, whether out of ignorance or intentionally. A fabricated hadith is not permissible to narrate except to warn against it and to clarify that it is fabricated; due to the Prophet's saying, peace and blessings be upon him: "Whoever intentionally fabricates a lie against me, let him take his seat in the Hellfire," and "Whoever narrates a hadith from me, believing it to be a lie, is one of the two liars [the narrator or the fabricator]."