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GH-36839: [CI][Docs] Update test-ubuntu-default-docs to use GitHub actions instead of Azure (#36840)

### Rationale for this change

Currently `test-ubuntu-default-docs` has been failing on Azure for the 13.0.0 RC0 and we had to use GitHub actions to generate the documentation.
Using the same base action for both preview-docs, test and packaging will improve maintainability.

### What changes are included in this PR?

Move `test-ubuntu-default-docs` to use GH actions instead of Azure.

### Are these changes tested?

Yes, with archery related tasks.

### Are there any user-facing changes?

No
* Closes: #36839

Authored-by: Raúl Cumplido <raulcumplido@gmail.com>
Signed-off-by: Sutou Kouhei <kou@clear-code.com>