Plugin manager improvements

https://github.com/jupyter/notebook/pull/7198 added the plugin manager to `left` area:

[notebook-pluginmanager.webm](https://github.com/jupyter/notebook/assets/591645/48857932-2b85-4d50-a761-63aa36fe20e1)

This enables the feature, which is still hidden by default. However this might not be the best in terms of UX, so we can probably iterate more on it.

Improvements to consider:

- open on the file browser page like the settings editor?
- add UI test

Start by reviewing PR #7198 and the plugin manager in the left area, then compare its behavior with the settings editor on the file browser page. Clarify the preferred placement and UI-test scope with maintainers; done means an agreed UX improvement is implemented and covered by a UI test.

Written by the indexing model from the issue text.