Region: Middle Earth
You know, I really agree with you about Finrod and I do enjoy that poem, though it is not my favourite.
I think (I say think, though I am very sure) my favourite is the Lament of the Rohirrim. Particularly this version of it: https://www.youtube.com/watch?v=XPrKSLpm0EY&ab_channel=AdeleMcAllister
Coming in second is Legolas' Song of the Sea, with some contest from The Fall of Gil-Galad and the Song of Durin
There are just so many! I like Treebeard's songs, the Road Goes Ever On (especially Pippin's version in the ROTK movie).
I could go on, but I've mentioned so many that I should probably let someone else chime in :)