“…There is no denying [the Tosefta] is a source of the Mishnah. There is also no denying that it is a commentary on some other text. So much of it cannot stand alone. But the text upon which it comments is not the Mishnah, as we know it today, but the urMishnah, a forerunner of the Mishnah. However, since so much of that early text entered into our Mishnah, with or without change, the Tosefta functions at one and the same time as the basis for many paragraphs of our Mishnah and as a commentary on many others."
Above: a fragment of the Codex Kaufman Mishna
The issue of which came first, the Mishna that we have today or the Tosefta, is a bit arcane for most of us. But the truth appears to be there was a proto-Mishna or original Mishna the Tosefta was a commentary to, and that the Tosefta serves as one of the main sources for the Mishna we have today – even though this isn't what you were likely taught in yeshiva.
Here's a journal article by Judith Hauptman that explains why the Tosefta is in all likelihood older than version of the Mishna we have today.
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The Tosefta is more liberal on issues of women's participation in public religious life – teaching halakha openly, even to male rabbis, and putting on tefillin, just to name two examples – than the Mishna we have today and the rabbinic tradition that follows it are.
What this shows is that there was an accepted rabbinic tradition at about the turn of the Common Era that was much more inclusive of women. Vestiges of it are seen in the Tosefta, and they're also seen in the practices (such as we know of them) of the first followers of Jesus and the Early Jerusalem Church at a time when they still kept halakha and where a few women played prominent roles.
Why was that more liberal rabbinic tradition quashed? I don't know.
But I do know that as rabbinic Judaism progressed, it blamed more and more of men's sins, natural disasters, wars and epidemics on women. Their "immodesty," the way their beauty supposedly causes men to sin, and their supposed "lightheadedness" all caused problems for the Jewish community that brought down divine retribution. Today, barely a month goes by without another haredi rabbi demanding that women's stockings get even thicker and more opaque, that their wigs get uglier or their dresses get even more drab and shapeless, all in order to save the community from some disaster, real or imagined, supposedly provoked by the sins men commit when women are not sufficiently burka-ized.
I think the natural tendency of the world is for the strong to oppress and exploit the weak, and societies have install strong safeguards and work very hard to prevent that from happening.
For whatever reason it happened, when rabbinic Judaism ceased doing so, when some of those safeguards were removed, women's religious roles shrank and their status dropped, and things like the following quote from a brita in the Talmud (Shabbat 152a) became the norm: "A woman is a vessel full of filth."
But that likely was not the view of the earliest rabbis, or at least a significant portion of them.
[Hat Tip: Shmuel.]