That's the whole point. Death is not there to last, most likely.
Now, who would get the immortality drugs? The rich and known, or random nobodies?
Same thing about today's reality - who would get rushed to their cryonics institution - someone no one would miss for months, or someone well known?
That's why it can matter.
If we are getting to the point of immortality drugs, it perfectly helps to my "conviction" (We are getting close to basement here, talking about something like religion): Take them. All. As many as you like. Just take them away from me.
What can not die, is not alive as well. And could never know how it is like to be alive.
One day will be my last. And everyday until then is precious. If it would not be a finite number of days, we would waste them. And sharing ideas, developing things together is what makes my day. One sudden day, I will die. But the ideas that I nurtured will live on. If I would hoard them - I would keep them for a future that might never come. Death will not ask me, if I need another week to get my business organized.
Also, I fear that, if the well-knowns get their immortality, the society will stagnate. New ideas are not adapted by old ideas getting proven wrong. They are adapted by the supporters of the old idea literally dying out.
Today, Magret Honecker died in her Exile in Chile. Guess what: In a society in your immortality world, she would have been among the first to get such a treatment. Staunchly on top, never realizing that her workers paradise was only a paradise for her, while millions of workers had to work for her like some ancient Egypt Pharaoh. She never realized her ideas had been wrong. She always thought everything else is wrong, but her ideas.
Who defines who deserves such a treatment, if not the "elite" that first of all thinks of itself as too important to not be immortal? Which in place means: Every other immortal will be more or less exactly like the initial elite and their children? And who will say "Follow our commandments, and maybe you will also be allowed to enter our circles."
Sorry, that kind of future sounds a lot like our past.