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:
Yea a label may be useful. Maybe it can be automatically added with some workflow since I currently give related PRs titles with a shared prefix?

Or alternatively I'm keeping track of all PRs towards the branch, related PRs towards main, and upstream issues (which you can find or redirect to from the very first comment). So essentially you can easily find everything related from the first comment in this issue, and maintainers can edit if more shortcuts are wanted.

Labels: Documentation