Update checks workflow to fix the broken link issue-3152

[Kushagra651]

Pull Request Description

Title: Update checks workflow and add CI badges to README

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Summary

This PR updates the CI/CD workflow (.github/workflows/checks.yml) and adds live status badges to README.md. It ensures broken links are automatically detected and provides real-time feedback on workflow health.

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Key Updates

1. Workflow Improvements

   • Added a broken links checker using [Lychee](https://lycheeverse.org/) with artifact upload (./lychee/out.md).
   • Runs after docs build (needs: check-docs) and integrates with existing jobs: check-docs, run-pylint, misspell, uv-lock.

2. README Enhancements

   • Added status badges for each CI job: Docs Build, Pylint, Spellcheck, UV Lock, Broken Links.
   • Badges link directly to workflow logs.

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Benefits

• Prevents broken links from being merged.
• Provides real-time CI feedback via badges.
• Improves visibility and maintainability of CI/CD workflows.

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Related Issues

• Fixes: #3152

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Reviewer Notes

• Verify the broken links job runs correctly.
• Check that badges render and workflow jobs function as expected.

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Signed Commits

• [x] All commits are signed.

[sgoggins]

ok ... i see the lint checker for broken links is working ... and that's the issue ...

[MoralCode]

Superseded by #3484