Fix Data Machine handler integration gate

[chubes4]

Summary

• Switches the Data Machine handler integration gate from class_exists() to trait_exists() for HandlerRegistrationTrait.
• Adds smoke coverage proving the GitHub issue publish handler bootstrap checks for the trait correctly.

Fixes #591.

Testing

• php tests/smoke-github-issue-publish-handler.php
• php tests/smoke-github-pr-publish-handler.php

AI assistance

• AI assistance: Yes
• Tool(s): OpenCode (GPT-5.5)
• Used for: Diagnosing the wp-site-generator loop failure, implementing the trait-gate fix, and drafting the PR text. Chris remains responsible for review and testing.