Death totally haunts this myth (pun intended). The story begins with Eurydice's fatal snake bite, continues with Orpheus' journey into the Underworld, and ends with Orpheus' murder at the hands of the Maenads. For a myth about escaping death, death sure feels pretty inescapable here.
There's a pretty famous theory that says there are five stages of grief: denial, anger, bargaining, depression, and acceptance. The myth of Orpheus and Eurydice spends a lot of time on stage 3: bargaining. When Orpheus meets Hades and Persephone, he convinces them to let Eurydice go, partly by talking about how, hey, everyone dies eventually, so why not let Eurydice live a little longer? They'll get her soul in the end, they should just be patient.
But despite successfully bargaining for Eurydice's release, Orpheus ends up losing his wife to death a second time. The lesson here might be that although love is powerful, it's ultimately not as powerful as death, which (as Orpheus told Hades) eventually conquers everything. Well that's kind of a downer.