It's a normal elision, yes, we all picture a public library when we say "library". But "free library" isn't redundant or weird, because "public" is a modifier of library, not a trait.

People tend to call their personal library a "book collection" or the like, but it's a library, in just the same way that a Little Free Library is.

So most people who read have at least a small private library, whether they think of it in those terms or not.