Improve performance of async_get_integration_with_requirements

[bdraco]

Proposed change

• Migrate to the asyncio.Future pattern instead of using asyncio.Event
• Use sets in a few places to avoid linear searching
• Check the cache when processing deps so we do not create tasks to process requirements for deps that have already been processed. (websocket_api, http, etc would get processed over and over again)

Type of change

• [ ] Dependency upgrade
• [ ] Bugfix (non-breaking change which fixes an issue)
• [ ] New integration (thank you!)
• [ ] New feature (which adds functionality to an existing integration)
• [ ] Deprecation (breaking change to happen in the future)
• [ ] Breaking change (fix/feature causing existing functionality to break)
• [x] Code quality improvements to existing code or addition of tests

Additional information

• This PR fixes or closes issue: fixes #
• This PR is related to issue:
• Link to documentation pull request:

Checklist

• [ ] The code change is tested and works locally.
• [ ] Local tests pass. Your PR cannot be merged unless tests pass
• [ ] There is no commented out code in this PR.
• [ ] I have followed the development checklist
• [ ] I have followed the perfect PR recommendations
• [ ] The code has been formatted using Ruff (ruff format homeassistant tests)
• [ ] Tests have been added to verify that the new code works.

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• [ ] Documentation added/updated for www.home-assistant.io

If the code communicates with devices, web services, or third-party tools:

• [ ] The manifest file has all fields filled out correctly.
  Updated and included derived files by running: python3 -m script.hassfest.
• [ ] New or updated dependencies have been added to requirements_all.txt.
  Updated by running python3 -m script.gen_requirements_all.
• [ ] For the updated dependencies - a link to the changelog, or at minimum a diff between library versions is added to the PR description.
• [ ] Untested files have been added to .coveragerc.

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