Post list: Search query should hide on navigation

[sirreal]

Steps to reproduce

1. Visit https://wordpress.com/posts/{siteSlug}
2. Click the search icon
3. Enter a query
4. Click "Switch site"
5. Pick any site, including the same one.
6. Notice that the search query remains visible but is not applied to the post list.

What I expected

The search query should be applied to the posts list. Alternatively, it should not be visible and the non-search bar should be shown.

What happened instead

Posts are shown unfiltered and the query is visible. This seems incoherent.

Screenshot / Video

Context / Source

Noticed while reviewing #18113

[rachelmcr]

Closing as a duplicate of #453.