Reconsider search bar and quick navigation

With #35010 (closed) and #36234 (closed) we added dropdowns to the Projects and Groups links in the top bar which allow you to quickly switch projects or groups. Both of these dropdowns contain search fields, which means we have three search fields inside the top bar.

This comment from https://gitlab.com/gitlab-org/gitlab-ce/issues/36234#note_61728113 outlines the problem and a potential solution:

However, the missing piece to solve here is the fact that right now we have 2 search inputs (global search and projects dropdown) and with this issue we would have 3 (we had already discussed this a while back). If we remove the search from the projects/groups dropdowns, the user would have to go to the global search to find a project/group that is not in the “Frequently visited” list. Now, we might think that having 3 search inputs is an issue, but the real UX issue is making the user go to different places just because we insist on having just 1 search input. So my vote here is to proceed and implement the groups dropdown with search (like the current projects dropdown).

Something that might solve this more holistically is implementing https://gitlab.com/gitlab-org/gitlab-ce/issues/36409 and, after that, changing the order of the navigation items from global to more specific:

  1. GitLab logo

  2. Search bar

  3. Projects, Groups, Activity, Milestones, Snippets

  4. Plus button

On the right side, we would only have personal items.

cc. @pedroms @jramsay

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