Honestly, that feels extreme. I never thought about it at the time back when I read LotR, but man, that feels wrong.
After all, humans have been dealing with this for as long as they've existed - whether outliving your parents (expected), outliving your spouse and friends (always possible), outliving your children (tragic), or even at the lowest end of that scale, outliving your pets (expected and tragic), outliving your loved ones is just something that will happen.
Tolkien deciding the elf side of a couple always dies of heartbreak because he dislikes the notion of the elf moving on and remarrying that much is kind of rubbish of Tolkien.
If parting is inevitable, then you want the other side to be happy that they were with you for as long as that was possible, and to keep being happy after you're gone.
Well first off, "moving on" when you've been together long enough becomes very difficult even for humans, for a being that's literally immortal the idea is probably excruciating, there have been many cases in history of this very thing happening with humans, much less in fantasy epics.
Also remember that the elves are already growing tired of their immortality, hence why they want to sail out of middle earth for the land of the gods, this is effectively death without the whole death part anyway (since it's heavily implied that this is where the souls of the dead go), because the elves have already seen their kind die in wars and felt loss, realizing that eternity is a VERY long time.
I should point out my grandfather very nearly went through this and only lived long enough to ensure that my cousin (which was in his care) grew up to the point he no longer needed him anymore, he even warned my father a few weeks beforehand that he should come visit him as "time was almost up".
My great grandfather's dog, which lived 33 years, died 3 days after my great grandfather died as well, simply put, not everyone can survive the loss of the one they consider their true companion.
My great-great grandmother lost the will to live when my great-great grandfather died and didn't live much longer either, i guess it's a somewhat common occurrence in my family.