Un-special-case search box on index page

[jsignell]

Related Issue(s):

• Closes Broken search on docs #1349

Description:

The search box was embedded in the page, but that was not obvious and made the topbar search behave strangely. This PR makes the index page the same as the other pages.

PR Checklist:

• pre-commit hooks pass locally
• Tests pass (run scripts/test)
• Documentation has been updated to reflect changes, if applicable
• This PR maintains or improves overall codebase code coverage.
• Changes are added to the CHANGELOG. See the docs for information about adding to the changelog.

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 91.07%. Comparing base (d9b354d) to head (d1635ec).

Additional details and impacted files

@@           Coverage Diff           @@
##             main    #1364   +/-   ##
=======================================
  Coverage   91.07%   91.07%           
=======================================
  Files          51       51           
  Lines        7070     7070           
  Branches     1012     1012           
=======================================
  Hits         6439     6439           
  Misses        451      451           
  Partials      180      180           

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