Fix auth controls for multi-spec API pages

[half-ogre]

Please check if the PR fulfills these requirements

• Followed the Contributing and Code of Conduct guidelines.
• Tests for the changes have been added (for bug fixes/features) with 100% code coverage.

What kind of change does this PR introduce? (Bug fix, feature, docs update, ...)

Bug fix.

This PR:

• Fixes the API page auth dropdown and cookie status badge not working after the multi-spec changes in v1.11.0
• The v1.11.0 template added {{@index}} suffixes to auth element IDs (e.g. api-auth-type-0) but api.js still referenced the unsuffixed IDs via getElementById
• Switches api.js to use class selectors (querySelector) instead of ID lookups, which works for both single and multi-spec configurations

[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 99.95%. Comparing base (78af1af) to head (98d71db).
⚠️ Report is 2 commits behind head on main.

Additional details and impacted files

@@           Coverage Diff           @@
##             main     #386   +/-   ##
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  Coverage   99.95%   99.95%           
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  Files          16       16           
  Lines        2364     2364           
  Branches      696      696           
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  Hits         2363     2363           
  Partials        1        1           

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[gemini-code-assist]

Code Review

This pull request updates the authentication logic in templates/modern/js/api.js by replacing ID-based element selection with class-based selectors. This modification ensures compatibility with multi-spec environments where element IDs are dynamically indexed. I have no feedback to provide.