What does transitive = true in Gradle exactly do (w.r.t. crashlytics)?

You are using the @aar notation.
It means that you want to download only the aar artifact, and no transitive dependencies.

You can check Dependency management in Gradle
in the official documentation. In particular:

An artifact only notation creates a module dependency which downloads only the artifact file with the specified extension. Existing module descriptors are ignored.

Using the @aar notation if you want to download the dependencies, you should add transitive=true.

I’d expect that omitting @aar it should work without adding the transitive attribute.