Brahmavadinis are women who learn the Vedas, recite mantras, and do Yajnas, as stated here.
How come Vedantic sampradayas don't permit women to do this?
I think it's because it's much easier to become a Sadyovadhu (housewife), and the duties of a housewife fulfill the same function as a Brahmavadini, as Manu has said:
2.67 - For females the Rites of marriage have been ordained to be their ‘Vedic Sacrament,’ the serving of the husbands their ‘residence with the Teacher,’ and the household-duties their ‘tending of fire.’
5.153 - There is no separate sacrificing for women, no observances, no fastings; it is by means of serving her husband that she becomes exalted in heaven.
The fact that Brahmavadinis and Sadyovadhus achieve the same result is also stated in Balambhatta's gloss of the Mitakshara:
And also stated in the Vishnu Purana by Vyasa:
Thus, eminent Brahmans, through these and other sources of anxiety, men attain their allotted spheres of Prajápati and the rest only by exceeding labour and suffering. This is not the case with women: a woman has only to honour her husband, in act, thought, and speech, to reach the same region to which he is elevated [heaven]; and she thus accomplishes her object without any great exertion.
Obviously, simply serving your husband to the best of your ability is much easier than learning the Vedas, memorizing it, begging for food, doing Yajnas, strict observance of austerities, etc.
The Harita smriti also says that in an "ancient age" (pura kalpe), women kept the sacred thread:
So, I think as humans started becoming more corrupt as time passed by, Manu and the Rishis gave women an easier option in the form of assisting their husbands.
And according to Nadadur Ammal, an ancient Sri Vaishnava acharya, in his work called the Yatilinga Samarthana, whenever there is a difficult option and an easy option that achieve the same results, as a rule, one would take the easy option:
However, Nadadur Ammal says "would not be undertaken," and not "should not be undertaken," because just because something is harder doesn't mean you're prohibited from taking that option.
But anyways, I think this is the reason why most women today are prohibited by the major Sampradayas from wearing the sacred thread.
On the other hand, men of the first 3 castes have no easier option; they are required to do the Upanayanam, recite the Vedas, etc.
But are there any Vedic scholars from any of the major Vedantic sampradayas today who have commented on this?