[Misc] Split up pooling tasks

[DarkLight1337]

Part of #10674

Expects #10827 to be merged first.

[github-actions]

👋 Hi! Thank you for contributing to the vLLM project.
Just a reminder: PRs would not trigger full CI run by default. Instead, it would only run fastcheck CI which starts running only a small and essential subset of CI tests to quickly catch errors. You can run other CI tests on top of those by going to your fastcheck build on Buildkite UI (linked in the PR checks section) and unblock them. If you do not have permission to unblock, ping simon-mo or khluu to add you in our Buildkite org.

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