DOC Update website to `pydata-sphinx-theme`

This issue is a continuation of [#26809](https://github.com/scikit-learn/scikit-learn/pull/26809) and aims to migrate the scikit-learn website towards [`pydata-sphinx-theme`](https://pydata-sphinx-theme.readthedocs.io/en/stable/). As this is an ambitious goal, this issue is to track the steps of the migration. cc @lucyleeow who guided me to open this issue.

#### Quick links

- [The `new_web_theme` branch](https://github.com/scikit-learn/scikit-learn/tree/new_web_theme)
- [Tracker for upstream issues](https://github.com/scikit-learn/scikit-learn/issues/28084#issuecomment-1893103722)
- Live preview: https://scikit-learn.org/_pst_preview

#### TODO before merging into `main`

- `doc-min-dependencies` is bypassed in CI for now and we need to reactivate it before merging into `main`. Also, we need to make sure all dependencies are documented, in particular `sphinx-design` which was not added in the very first setup PR. See also [#28379](https://github.com/scikit-learn/scikit-learn/pull/28379).
- Remove the `new_web_theme` part in `.circleci/config.yml`.
- In `conf.py`, change the version switcher link to `https://scikit-learn.org/dev/_static/versions.json`.
- Remove `themes/` and update `exclude_patterns` in `conf.py`: these are only useful for the old theme.
- Pin higher versions of `sphinx`, `pydata-sphinx-theme`, and `sphinx-gallery`. See also [tracker for upstream issues](https://github.com/scikit-learn/scikit-learn/issues/28084#issuecomment-1893103722).

#### Tracking work towards the `new_web_theme` branch

- [x] https://github.com/scikit-learn/scikit-learn/pull/28132 
- [x] https://github.com/scikit-learn/scikit-learn/pull/28331
- [x] https://github.com/scikit-learn/scikit-learn/pull/28336
- [x] https://github.com/scikit-learn/scikit-learn/pull/28347
- [x] https://github.com/scikit-learn/scikit-learn/pull/28353
- [x] https://github.com/scikit-learn/scikit-learn/pull/28379
- [x] https://github.com/scikit-learn/scikit-learn/pull/28401
- [x] https://github.com/scikit-learn/scikit-learn/pull/28409
- [x] https://github.com/scikit-learn/scikit-learn/pull/28408
- [x] https://github.com/scikit-learn/scikit-learn/pull/28415
- [x] https://github.com/scikit-learn/scikit-learn/pull/28428
- [x] https://github.com/scikit-learn/scikit-learn/pull/28448
- [ ] https://github.com/scikit-learn/scikit-learn/pull/28478
- [x] https://github.com/scikit-learn/scikit-learn/pull/28511
- [x] https://github.com/scikit-learn/scikit-learn/pull/28512
- [x] https://github.com/scikit-learn/scikit-learn/pull/28623
- [x] https://github.com/scikit-learn/scikit-learn/pull/28630
- [ ] Fix admonitions (**Todo:** make e.g. `.. note:: something` into `.. admonition:: something` with `:class: note`)

#### Tracking work related by towards the `main` branch

- [x] https://github.com/scikit-learn/scikit-learn/pull/28387
- [x] https://github.com/scikit-learn/scikit-learn/pull/28376
- [x] https://github.com/scikit-learn/scikit-learn/pull/28315
- [x] https://github.com/scikit-learn/scikit-learn/pull/28272
- [x] https://github.com/scikit-learn/scikit-learn/pull/28271
- [x] https://github.com/scikit-learn/scikit-learn/pull/28255
- [x] https://github.com/scikit-learn/scikit-learn/pull/28134
- [x] https://github.com/scikit-learn/scikit-learn/pull/28107

COMMENT:
Should we add a label for PRs related to this effort? I'm generally not the biggest fan of labels, but it seems like it could be useful here to filter PRs to just those related to this issue. WDYT?

Labels: Documentation