I think all humanity will benefit from making certain advantageous people immortal. Just imagine what immortal Einstein, Stephen Hawking, or Elon Musk could achieve?
I certainly don't think we should give immortality to anyone who wishes it. Just look at how many useless people there are. But, if we make useful people immortal, than slowly, over time, the percentage of useful and useless people will shift. Imagine if countless genius immortal brains are working day and night to cure cancer, to invent warp drive or quantum computers, instead of smoking pot in mom's basement. If I, personally, will not be judged as worthy of immortality, I will gladly give up my life for the greater good.
You might say I'm borderline with faschism, but I disagree. I'm totally against harming or eliminating anyone using force.
Now, as to countering overpopulation, I think if we achieve immortality, birth rates should be strictly controlled. I do not believe that unborn human equals non-existent human (I argued about that extensively in couple of other topics some months back). So we should only make babies to accommodate available resources, and only then. It's much more important to achieve happiness for everyone, even if it means lowering the overall population numbers.