No way to access the builder in directives

I was asked in https://github.com/sphinx-doc/sphinx/issues/13072#issuecomment-3594904760 to open a new issue for this

Describe the bug

With the new logic discussed in #13072 I don't see a way how a directive can access the builder, to access e.g. app.builder.outdir and app.builder.name. The app can currently be fetched over self.state.document.settings.env.app which won't then work anymore.

You can find my code here: https://github.com/SAP/swagger-plugin-for-sphinx

How to Reproduce

n/a

Environment Information

Platform:              linux; (Linux-5.14.21-150500.55.124-default-x86_64-with-glibc2.31)
Python version:        3.14.0 (main, Nov 10 2025, 09:33:32) [GCC 7.5.0])
Python implementation: CPython
Sphinx version:        9.0.0
Docutils version:      0.21.2
Jinja2 version:        3.1.6
Pygments version:      2.19.2

Start by reading the discussion in #13072 and the directive usage in the linked swagger-plugin-for-sphinx project; the issue does not name a Sphinx source file or test. Determine how directives should access the builder, including its outdir and name, and add coverage showing that this access works with the new logic.