CI: rework workflow to leverage caching

In order to shorten the CI/CD process when including new examples, it could be interesting rework the way we are doing things by building the documentation while only executing the new example(s).

Idea of how to:
- cache previous examples
- use `nbsphinx_execute="auto"`

**Not sure that we should go with that approach yet**. We should wait to see how this repo evolves before doing anything as proposed.

Another solution of CI/CD process could be to add a job to create a sub documentation containing only the new examples and to proceed with the full documentation if this job works.

No specific files or tests are named. Review the repository's CI workflow and documentation build entry points, then compare caching previous examples with using nbsphinx_execute="auto" and the alternative sub-documentation job; the work is done only after the repository has chosen and validated a concrete approach.

Written by the indexing model from the issue text.