Why can’t sudras read vedas according to smritis?
I have used this YajurVeda mantra in more than one answers of mine till date. One such answer is this one.
The correct translation, word to word, is as follows:
YathA imAm vAcham kalyAnim AvadAni janebhyah (1)
BrahmarAjanyAbhyAm sudrAya cha AryAya cha (2)
SwAya cha aranAya cha (3)
May i speak the sacred word to the masses of the people (janebhya) (1) to the brahmana, kshatriya, to the sudra and the Arya (2) and to our own men and the strangers (3).
Shukla Yajur Veda 26.2
So, what it says is that the sacred words (or the Veda mantras) can be spoken to everyone, irrespective of their castes, even if he is a stranger (probably a Mleccha or one who is alien to the Vedic culture is implied by this word).
So, everyone can hear Veda mantras. But it does not say anywhere that everyone can recite or learn them too.
What the Smritis and the Puranas say is that a Sudra by birth does not even have a right to Gayatri initiation (which is what that makes one competent for Veda study).
So, if the Vedas say that even Sudras can recite Vedas, they have right to Vedic initiation, only then contradiction between Sruti-Smriti will arise. But, as of now there are no such contradictions. Even the Agamas say that for Sudra , there is no Upanayana.
But, if we find verses from Smritis which say that Sudras should not hear the Vedas, then there is a clear contradiction between Sruti-Smriti and we might reject them.
Another thing to be taken into consideration is the fact that a huge amount of scriptures (including the Vedas) have been lost. And we can't be sure what's in those lost portions. So, we have to come to a conclusion after consulting all the scriptures that are available with us now.
And, not only for the Vedas, even for the Puranas, the rule is that the Sudras should not read them by themselves. The Brahmins must recite and they should only hear the recitation.
So for the satisfaction of one’s own S’akti, all can read this, without contradicting each other. Never any woman nor any S’ûdra, is to read this herself or himself, even out of ignorance; rather they should hear this from the mouth of a Brâhmana. This is the rule of the S’âstras.
From Devi Bhagavatam's last chapter.
Updating the answer:
Yet another relevant mantra from the Atharva Veda (AV) quite clearly mentions that the mantra (or the sacred word) can be spoken to the masses.
AsvinA sAradhana mA madhunAnka shubhaspati (1) YathA bhargasvatim vAcha AvadAni janAm anu (2)
Asvins, Lords of light, fill me with sweetness of the bee-honey (1), so may i speak the glorious Word to the masses of the people (2).
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