You almost certainly want to build the app inside a container, but probably not the same container that result from the build.

[ed: well maybe not for a simple go project, as go has great cross compilation support. But if there's a bit of Javascript/(s)css/static resource munging - a container with a predictable build environment might become appealing. ]

Isn't this the purpose of multi stage builds? To have one stage for building it and another, from scratch, to run it?

It is one of the purposes, yes. The idea being that you don't need build artifacts in containers which run the application.