This scene reminds me of the one in The Tale of Aragorn and Arwen when Aragorn dies. I can't quote from the book right now, but Arwen says soething like this: "Not until now did I understand in full measure the bitterness of the fate of Men". Even though this is a different kind of relationship than a mother---daughter one, it is still a loosing a loved one. Could Melian perhaps be experiencing something similar? Maybe she also took for granted that Men die, and she might have grieved a bit for the death of some of them (not that she knew any of them close until Beren came). She probably felt sympathetic towards their losses, and as much as she could empathetic, but the real thing was unknown to her until Luthien left her.
And, like
Bęthberry said, she was unprepared for this, not only because she was immortal, but also because Luthien was the first immortal to marry a man. No such a thing existed before for elian to take comfort in.
But Melian has noone to blame but herself for Luthien's choice. Luthien followed her other's steps - Melian was first maia to marry a non-ainur, and Luthien the first elf to marry a man. Melian passed it on, didn't she?
*I didn't really mean that. I was just joking.
*