The Mishnah was redacted around 200 CE, more than a century after the Temple was destroyed, and it is the only detailed source for most of the procedure. Josephus, Philo and the Qumran texts corroborate the institutions but very little of the fine detail, and the two bodies of evidence were produced for different purposes.
What would resolve this A systematic detail-by-detail comparison of Mishnaic procedure against the pre-70 evidence, distinguishing points corroborated by Josephus, Philo, Qumran and the surviving epigraphy from points attested only in the Mishnah, would show where the account is anchored and where it is not. Fresh documentary or epigraphic material from the Temple period would move individual points.
OpenRabbinics, Second Temple history, epigraphyShaye J. D. Cohen, From the Maccabees to the Mishnah