According to the four Sunni schools, it is permissible to follow any school of thought (madhhab) provided that:
- The sources are the Qur'an or the authenticated Sunnah (depending on the extent of the ruling and its impact).
- If their school of thought is in contradiction with Qur'an or Sunnah, either through misinterpretation of a verse, misunderstanding, or lack of knowledge that a hadith exists to address a specific situation, all four scholars explicitly instructed that their teachings be ignored, and to adopt the school of thought that provides better evidence.
- All four agreed that if someone quotes them to quote their evidence, too. All agreed that they should not be followed, but to follow their evidence instead, i.e., their opinions are to be evidenced for from the Qur'an and Sunnah and not to be used as evidence of and by itself.
None of the scholars — neither from the four widely-accepted madhhabs, nor from the extinct madhaabs — started by documenting their fatwas and labeling said documentation as the basis or foundation of a madhhab. Typically, all madhaabs started with scholars that have a high level of knowledge of the religion's different aspects; books that these scholars wrote; a number of students that further documented, compiled, and explained their fatwas; then judges that adopted their school of thought, and used it in court.
For example, Al-Awza'i (one of the schools you mentioned, which is quite similar to Al-Shafei madhaab) was adopted in the courts of Al-Sham (Levant) where he was from (until Abu Zar'a Muhammad ibn 'Uthman introduced the Shafei madhaab there), and in Al-Andalus (introduced by his follower Sa'sa'a ibn Salam) for 220 years, then it was abandoned as judges at the time stopped using it as a reference to use other schools (as mentioned by Abu Al-Wafa' ibn 'Aqil), and very few references survived (as mentioned by Ibn Badran), none of which constitutes a comprehensive reference to the madhaab, merely quoting it in general literature on jurisprudence. Refer to To'haft Al-Anam (p. 82) by Al-Fakhoury (تحفة الأنام مختصر تاريخ الإسلام) and Al-Madkhal Ela Derasat Al-Madares (المدخل إلى دراسة المدارس والمذاهب الفقهية) for more information.
As for your quote about Al-Dhahiriya school of thought, the material for this school of thought still exists and is often referenced by followers of other schools of thought. The topic of why Al-Dhahiriya is not a madhaab in use has been widely debated and discussed. Mostly, outcome is opinion-based. The reasons mentioned revolve around the low number of students of Ibn Hazm (hence, lower number of books, hence lower level of adoption by other scholars and judges) due to:
- Ibn Hazm's use of strong words in criticizing other scholars' views, which drove a number of people away from his views.
- His students did not document his works due to his verbiage, so the only books available today are the ones written by Ibn Hazm himself.
- Lack of confidence expressed by several major scholars in Ibn Hazm's works due to his acceptance of only explicit wording (Dhahir Al-Qawl), rejection of un-evidenced concession (ijmaa') and conceptual deduction (qiyas jali), in addition to his occasional and controversial use of philosophical views and materials, which resulted in some contradictions in his fatwas.
Generally speaking, though, Ibn Hazm was very similar to the other four madhhabs in following the salaf school of thought, but the spreading of a madhhab is highly dependent on the confidence scholars show in that school of thought,
and its use in the court system.
For the sake of documenting some of the extinct madhaabs, here is a commonly referred-to list:
- 'Abdullah ibn Mas'ud
- 'Aisha bint Abu Bakr
- 'Abdullah ibn 'Umar
- 'Umar ibn 'Abdel-Aziz
- Mujahid ibn Jabr
- 'Aamer ibn Shorahbil
- 'Ata' ibn Abi Rabah
- Sufyan Al-Thawri
- Al-Layth ibn Sa'd
- Sufyan ibn 'Uyaynah
- Is'haq ibn Rahwayh
- Dawud Al-Dhahiri
- Ibn Jarir Al-Ṭabari
While considered extinct madhaabs, almost all of them are widely quoted in textbooks, and are used by scholars and followers of the four schools of thought even nowadays.