Revert Algolia proxy changes

[christian-byrne]

Reverts the Algolia proxy configuration changes from #4030 and #4058, restoring the direct Algolia client connection.

Fixes issues with the proxy configuration by reverting to the original implementation.

┆Issue is synchronized with this Notion page by Unito

[christian-byrne]

Unrelated test errors likely due to upstream changes to templates (e.g., order, thumbnail image adjustments)

  2 failed
    [chromium] › browser_tests/tests/templates.spec.ts:148:3 › Templates › template cards are dynamically sized and responsive 
    [chromium] › browser_tests/tests/templates.spec.ts:176:3 › Templates › hover effects work on template cards 

[webfiltered]

Ignoring test failure - unrelated and can't fix now.