The Gollum backstory is actually in the Lord of the Rings - it's the prologue to it. It's that animated film, not the Hobbit or Return. (And no, you aren't inventing it - it's a scene with silhouettes or shadows against a red background, fascinating animation technique and at the very begging or narrative exposition of the animated take on Lord of the Rings.) I know this because I started watching the animated take on Lord of the Rings - and ran smack dab into it. It was a pleasant surprise I had forgotten all about it. (Small wonder - I've not seen the film since the 1980s-90s. I saw it on video, I think. )

Interesting - you loved the "writing" of the Hobbit. I loved the story...I can't remember if I loved the writing or not? I read it in 1977-80 or thereabouts. I do remember it feeling rather easy to read, and captivating.