Yes, indeed.
Urwen, your beginning was so close because Sirion is geographically born right next to Mithrim. And you named a couple things Hui's explanation did not, albeit, as I said before, not their intended places. So here is the riddle as I was thinking about it:
I was born in place up high - Eithel Sirion, the Wethrin mountains
Near growth turned to dust, and steel to rust. - near Anfauglith and Haudh-en-Nirnaeth
And long ahead my road did lie:
I passed the wall, I hid from all. - Andram, and disappears underground after the Falls of Sirion
And many met me on my way:
The one that came by many caves, Esgalduin by Menegroth (though Narog fits as well)
The one that washed the grief away, - Teiglin, which carried away Nienor
The one that's dead upon hard bed. - the Dry River
I always yearned towards the sea, - and always flows to it
Till in the war I was no more. - War of Wrath
But while I was, I lead with me
A helper great who aided fate. - Ulmo would channel his power through Sirion to help out, e.g. to hide Turgon's retreat from the Nirnaeth
And without further ado, over to Hui!