Yes, I have a given Hindu name (Amar Nath - associated with the premier Kashmir Shiva shrine). Generally the honor of choosing the name is given to the grandfather or equivalent elder at one's home, twelfth day after birth or so depending upon the community, done with feasting of close relatives. People also were given a secret name according to the birth chart at one time. It is for personal and religious identification. Used on all civil and legal occasions. I have chosen a Vedic nickname for use in forums.
Sanskara (rite of passage) - Wikipedia
Amarnath cave and shrine, Jammu & Kashmir, 3,888 m (12,756 ft)
They would gladly do it for any person. The idea is to bind the person with their sect. They also have quick marriage ceremonies for people in a hurry and the ritual includes chanting the Gayatri mantra thrice in the beginning and thrice at the end of it. That is about all. The main-line Hindu priests will also do it but their procedure is longer and more costly. However, as you have pointed out, none is absolutely essential and all depends on the choice of the person.
How are the judges concerned about any one's name. I do not know which country you live in. Do the judges, then, ask the person to change his/her name?
In India, name changes is no problem other than paper work - advertisements in two newspapers to see if anyone has objections, and an affidavit to a magistrate. Then change of name in all legal papers. That takes time and effort.
Actually, my son is in the process of changing the selling of my grandson's name, from '
Aaradhya' to '
Aradhya'. I had told my daughter-in-law that this spelling is ridiculous. They have woken up to it after 17 years. That means a spelling change in the passport, school records, bank account and Aadhar card (Indian government personal identification). I do not know how far they have done it.