[FLINK-40165][ci] Add docs check to Github Actions CI stage

[dalelane]

What is the purpose of the change

Adding a new workflow to make sure docs aren't broken as part of the Github Actions CI pipeline for pull requests.

The goal is to achieve parity with the docs_404_check job that is included in the ci stage of the Azure Pipelines
azure-pipelines.yml which I'm treating as a reference implementation.

Brief change log

Added a new docs check workflow, based on the Azure Pipelines equivalent.

Verifying this change

Manually triggered the Github Actions pipeline in my own fork

(see dalelane#2)

Does this pull request potentially affect one of the following parts:

• Dependencies (does it add or upgrade a dependency): no
• The public API, i.e., is any changed class annotated with @Public(Evolving): no
• The serializers: no
• The runtime per-record code paths (performance sensitive): no
• Anything that affects deployment or recovery: JobManager (and its components), Checkpointing, Kubernetes/Yarn, ZooKeeper: no
• The S3 file system connector: no

Documentation

• Does this pull request introduce a new feature? no
• If yes, how is the feature documented? not applicable

Was generative AI tooling used to co-author this PR?

no

[flinkbot]

CI report:

• 597c12e Azure: FAILURE

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• @flinkbot run azure re-run the last Azure build