Add desktop content switcher to the home page

[obulat]

Fixes

Related to #592

Description

This PR adds a Content Type switcher to the desktop version of the homepage. I've done it to test out how well the content switcher popover works when not in the header, and to see how best to handle state. It turns out that to make the component work well both in the header and on the homepage, it is easier to handle state (i.e. currently selected item) of the VContentTypePopover from the parent.

Testing Instructions

Run pnpm run dev, and open the home page. You should be able to see the content switcher inside the search bar, and also to select a content type from the dropdown (popover). The search itself is not functioning yet.

Checklist

• My pull request has a descriptive title (not a vague title like Update index.md).
• My pull request targets the default branch of the repository (main) or a parent feature branch.
• My commit messages follow best practices.
• My code follows the established code style of the repository.
• I added or updated tests for the changes I made (if applicable).
• I added or updated documentation (if applicable).
• I tried running the project locally and verified that there are no visible errors.

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[sarayourfriend]

How does this PR related to #607 ?

[sarayourfriend]

I always liked this approach to mapping variables to a different result (using an object inline). 🙂

[obulat]

How does this PR related to #607 ?

I have been thinking about the Content Switcher only as part of the Header for a very long time. And just a couple of days ago I looked at the homepage, and realized that the same component is used inside the search bar. This made me rethink the organization of all of the components related to the switching of content types.
I will use the changes from this PR in #607.

[zackkrida]

Oh, what an amazing surprise, I assumed this would be on the old homepage! This is great. @dhruvkb I want to make sure you're aware of this since you're working on finishing the homepage.

[zackkrida]

@dhruvkb feel free to merge this without additional review if it is helpful to your homepage PR; the specific details can be discussed there.

[dhruvkb]

It will definitely be very useful, let me resolve the conflicts and merge it. Tagging @obulat for the notification.