Why is the narrator the one leaving in "Streets of Gold"?

The narrator is leaving to let his loved one die peacefully

_{For the purposes of this answer, “narrator” refers to the singer and “loved one” refers to the subject of the song.}

The narrator knows that his loved one isn’t afraid of death.

I don't remember seeing fear in your eyes
When you were fading
The day we said our goodbyes

His loved one is ready to die - there is no "fear in [their] eyes" while they are "fading" away or saying "goodbyes". The narrator sings about his difficulty coming to terms with this inevitability:

It's easy to say that there's a reason for this
Much harder to know
That what we say is true
Everything we hold could someday slip away

The narrator acknowledges that simply saying “there’s a reason for this” is easy to say but harder to take comfort in. It’s difficult to believe that such comforts are true when a loved one is dying in front of you. He realizes how “[e]verything we hold could someday slip away”.

And yet, even though letting go is hard, the narrator sings about how he believes his loved one will go to a better place:

I've got a feeling I'm gonna find you again
Just in a place where love can’t die

So the narrator resolves not to try to keep his loved one in this world. He realizes that even though it is hard, he must let his loved one go. Therefore he is “leaving to let [them] go” . This is explicitly outlined in the chorus:

I want you to know
I'm leaving to let you go
And someday we'll walk upon
The streets of gold

He is leaving, allowing his loved one to go to a better place. “Someday” the narrator will join them there, in that better place with “streets of gold”.

Therefore, the narrator specifically is leaving to allow his loved one to pass peacefully to a better place.

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