You gave the example of Śudra–Rathakār from Baudhāyana Grihya Sutra in which his education can be initiated in rainy season and in the very next line you mentioned Śudra–Rathakār as an offspring of a Vaiśya male and Śudra female. As in Indian tradition, mostly a child is known by his father's name. So, do you think here in Baudhāyana Grihya Sutra Śudra–Rathakār was allowed for formal education only because of his father lineage?
As far as I know, Manusmriti had very different views on that. Manusmriti says " If a Brahman or any upper cast, who marries a Shudra woman, degrades himself and his whole family, becomes morally degenerated, loses Brahman status and his children too will attain status of shudra." But what you are saying may be possible, because later Vedic period diluted Manusmriti.
Remember Manusmriti has changed versions.
In Vedic or Indian traditions woman is always given importance. Therefore, if a lower varna woman is married to a upper varna man ,her varna will be the varna of her husband. Same will be with her children.
My comment is limited to how a child was known, in ancient times.
It is not true that child was mostly known via his/her father's name.
For example I will quote names of few personalities, referred after their mothers….and this does not appear to be exception at all.
Recently there was a discussion on social media, in some context (possibly gender) and this subject came up and numerous individuals quoted such names (backed by specific shlok) and the list was very long.
The real question is if there is a systematic description in the texts what will be the progeny if parents are from different varnas. The second question is which one is older – Bodhayana grihya sutra or Manusmriti?
Another point on the side is how the Indian constitution accepts marriage between so called upper caste and SC/ST or OBC to decide the reservation?
What I know Manusmriti describes something about progeny. But in the Vedic text, we need to check.
Manusmriti should be oldest. It's real date is unknown. However, Bodhayana Grihya Sutra is compiled between 7th to 8th century BCE.
Indian Constitution or Hindu Marriage Act, 1955 does not mention this. They only mentioned about criteria of marriage and they allowed marriage in sapindas (five generation in case of the father and three generation in case of the mother). But this question comes in front of the full bench of Bombay High Court in 2010. In 2012, Supreme Court of India also endorsed the decision given by Bombay High Court.
Bodhayana Grihyasutra is Vedanga / later vedic texts ,while Manusmriti is Smriti . so chronologically Manusmriti is later .
Very informative article. Sir, I want to read more things about Vedic education. Ashram system. People's civic life. Value system during that periods.