DOC: migrate from sphinx-panels to sphinx-design
- closes #52939

Eventually it is the goal that the sphinx-design package replaces the sphinx-panels package (see https://github.com/executablebooks/sphinx-panels/issues/67 and https://sphinx-design.readthedocs.io/en/furo-theme/get_started.html#migrating-from-sphinx-panels, sphinx-design has the same (and more) features, but with a refactored and more flexible code base). 

So this PR moves our usage of sphinx-panels to sphinx-design (at least already for the front page, still need to do the same for the getting started pages)

It gives almost identical result as we have now (screenshot is with this PR, compare with eg https://pandas.pydata.org/pandas-docs/version/1.3/):

![image](https://user-images.githubusercontent.com/1020496/156773002-735b352e-26ad-408e-b792-9f3325f2e277.png)

looks like failing doc build ```
Extension error:
Could not import extension sphinx_design (exception: No module named 'sphinx_design')
```
Is a dependency missing? Yes, I didn't yet update the CI / requirements, because there is also more sphinx-panels usage to convert. Should have made it a draft PR! This pull request is stale because it has been open for thirty days with no activity. Please [update](https://pandas.pydata.org/pandas-docs/stable/development/contributing.html#updating-your-pull-request) and respond to this comment if you're still interested in working on this. is this active? > is this active?

thanks @jorisvandenbossche for the additional commits. Is this ready for review? Closing to clear the queue, but feel free to reopen if you have time and interest to work on it again.  /preview Website preview of this PR available at: https://pandas.pydata.org/preview/46227/ /preview Website preview of this PR available at: https://pandas.pydata.org/preview/46227/ /preview Website preview of this PR available at: https://pandas.pydata.org/preview/46227/ @phofl @datapythonista @jorisvandenbossche I think this PR is ready now if you could review when you have time /preview Website preview of this PR available at: https://pandas.pydata.org/preview/46227/