"termination of this round"...sounds like reincarnation belief. have you questioned this? is it real, knowable or just unverified belief?
as for your question: "why we generally place so much importance on the recognition of the lives of the famous?"
well, generally, people are famous because they bring others pleasure. for example, the surfer. he was an incredible surfer for those who appreciate surfing. plus he was very a likeable guy
separation from what is loved and pleasing brings recognition and saddness
if a human being touches one's life, then there will be recognition when they pass away
it is the same as a mother or father. when they pass away, this affects those who are close to them
when a mere stranger passes away, naturally that will not affect one because there is no intimacy, love or attachment there
it is all plain natural psychological science really
nothing to do with reincarnation
even animals feel sorrow when they are separated from their loved ones