I'm a Hindu and I have never heard of the system you described. All Hindus I know of inherit their father's LAST name, not their first name.
https://www.telegraphindia.com/opinion/what-s-in-a-name-ofte...
so wheras in the North, you'd have Narayan Sharma's son being named Vishal Sharma, in Tamil Nadu Sriram Narayan's son would be Vishal Sriram.
So yes, the system does exist
Due Disclosure: South Indian, not a Brahmin, but I have many friends whose names follow this system.
Caste does not say your ancestors did that job, caste meant a person from a particular caste could only do that job and could not take up other jobs, people from other castes could not take their jobs and were disqualified because you were not born in a particular list of families. Some castes were prohibited from getting an education and were killed for this.