gcd is so ubiquitous, and frequently used, and part of every
other language I have regularly used
I've never needed it.
Or to be a little less glib about it, your experience of programming is not the entirety of programming, and in many domains, gcd is either not relevant at all, or only relevant in certain subdomains that are already wrapped away in more directly relevant libraries. I've never needed gcd, but I hardly ever start a project that doesn't need to parse JWTs.
You also make a direct correlation between "developer happiness" and "available out of the box." As a long time python developer, I'm rarely less happy than when I'm trying to connect to a web server using the "available out of the box" urllib. Requests is better, easier to use, readily available, and known to basically everyone in the community who deals with HTTP services. The python standard library is full of cruft that nobody uses or wants to use anymore.
Why is Option in std, and not gcd? Option is part of the public interface of most rust projects. If there were multiple implementations of Option, you'd need converter functions to get from type to type when dealing with multiple crates. If you have six dependencies that each use a different implementation of GCD, you won't notice or care, as long as they're all correct.
You found a community of people who help with rust problems. All you had to do was say "hey fam, is there an easy way to do a gcd?" Then you know about the crate, and your developer happiness continues.
Don't get me wrong. I hear you. You like batteries-included standard libraries. Rust doesn't have one, and it's not going to have one. Not because of ideological purity, but because we've seen the problems they cause, and don't want our language to go down that road. If that's a dealbreaker for you, that's okay. There are other languages out there. But maybe give rust a little more of a shot. I think you'll find that the community is very thoughtful, and the bulk of the design decisions the community has made have made for a very positive developer experience.