ci: clean up doc CI jobs

[ClementTsang]

Description

A description of the change, what it does, and why it was made. If relevant (e.g. UI changes), please also provide screenshots/recordings:

Unify doc deployment workflows in release and nightly.

Issue

If applicable, what issue does this address?

Closes: #

Testing

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If this change affects the program, please also indicate which platforms were tested:

• Windows
• macOS (specify version below)
• Linux (specify distro below)
• Other (specify below)

Checklist

Ensure all of these are met:

• If this PR adds or changes a dependency, please justify this in the description
• If this is a code change, areas your change affects have been linted using (cargo fmt)
• If this is a code change, your changes pass cargo clippy --all -- -D warnings
• If this is a code change, new tests were added if relevant
• If this is a code change, your changes pass cargo test
• The change has been tested to work (see above) and doesn't appear to break other things
• Documentation has been updated if needed (README.md, help menu, docs, configs, etc.)
• There are no merge conflicts
• You have reviewed your changes first
• The pull request passes the provided CI pipeline

Other

Anything else that maintainers should know about this PR: