He lives soooo long. He has to know and accept that his companions don't.
Except the Ponds were far more than his normal companions. They were his family. By the end of it, Amy and Rory had pretty much become his adopted parents (age differences notwithstanding), and then they were ripped
out of his life. I thought his angst was perfectly justified in this instance.
I grow a bit tired of every companion being designed as the most special companion ever. Like what you said with the Ponds. Rose was supposed to be that. Clara is also supposed to be that having saved every incarnation of the Doctor. Etc. Gets old.
And, it forces the contrived exits that were mentioned in this thread. If such and such a companion is so amazingly important to the Doctor, they wouldn't leave under normal circumstances so some contrived circumstances have to be created to force them to leave and stay separated from the Doctor (ala Rose, Donna, Amy, Rory all have had reasons that forced them to stay away from the Doctor).