Dear Nandani in fact Sutas could have also become Kathavachak. The entire story of Mahabharat as we know is Utshrava a Sut telling to sage Shaunak. And it is mentioned that when Sut Utshrava was narrating this story he was given a higher seat than the sages there.
Actually if you read any katha, it would start as "Sut ji kehte hain" or suta says, this clearly shows that the Sutas were the kathavachaks.
I personally feel that since Sutas were the children of a Brahmin mother and Kshatriya father, so they had an option of choosing either of the professions but in a very lower order, like being a low rank soldier in an army (or may be charioteer) or a religious story teller (which is essential part of any Puja but the Brahmins do much more)
Had being a charioteer such a menial job Sri Krishna and king Shalya would have never agreed to it. (The former would not have been asked by Arjun, and the latter would have never agreed how so much ever he was indebted by the Athithi satkar)
You are right in saying that caste system had become stringent by then, but it was only till the choice of profession, the untouchability had not entered the society by then.
I don't remember the exact verse, but there is a mention in Mahabharat about the Shudra sevaks serving food to the guests at the palace of Yudhishthir.
Kyunki Saas Bhi Kabhi Bahu Thi 2
Anupamaa
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