[Feature]: Exchange link checking action with a maintained one

[theBadT]

Problem Statement

https://github.com/gaurav-nelson/github-action-markdown-link-check has been deprecated by 04/ 2025 but is still being used inside our template workflows. This causes uncertainty at the users side.

Proposed Solution

Exchange link checking with https://github.com/lycheeverse/lychee-action

Alternatives Considered

No response

Additional Context

No response

Code of Conduct

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