[Bugfix] Enable API server headless mode and scale-out using Dockerfile entrypoint

[DarkLight1337]

Purpose

I found that API server headless mode and scale-out is silently ignored using the Dockerfile's entrypoint (python -m vllm.entrypoints.openai.api_server) because the logic for applying API server scale-out is only run inside the CLI file. So, I have moved the relevant logic to the main api_server.py to make them available to both entrypoints.

cc @njhill

Test Plan

Test Result

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• The purpose of the PR, such as "Fix some issue (link existing issues this PR will resolve)".
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[gemini-code-assist]

Code Review

This pull request refactors the API server startup logic to fix an issue where scaling out the API server did not work with the Dockerfile entrypoint. The core logic for handling different server modes (single, multi-process, headless) has been moved from the CLI-specific file vllm/entrypoints/cli/serve.py to the more general vllm/entrypoints/openai/api_server.py. This allows both the CLI entrypoint and the direct module entrypoint (python -m ...) to use the same startup logic. The changes are well-structured, and the refactoring is clean. The related test and comments have also been updated accordingly. The fix correctly addresses the described problem.

[DarkLight1337]

Closing as superseded by #25967