Remove task header launcher menus

[k11kirky]

Problem

Task headers include unwanted launchers for external IDEs and canned PostHog prompts.

Changes

• Remove the external app opener from task and skill headers
• Remove the task-header quick prompt menu and its persisted state
• Remove associated shortcuts, analytics, stories, and dead component code

Why: Task header actions should stay focused on the active task without launching local IDEs or injecting canned follow-up prompts.

How did you test this?

• pnpm typecheck --filter=@posthog/ui
• pnpm --filter @posthog/ui test (2,407 tests passed)
• pnpm exec biome check on changed source files
• Searched for removed component, store, shortcut, and analytics references

Automatic notifications

• Publish to changelog?
• Alert Sales and Marketing teams?

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Created with PostHog Code

[trunk-io]

😎 Merged directly without going through the merge queue, as the queue was empty and the PR was up to date with the target branch - details.

[github-actions]

React Doctor found no issues in the changed files. 🎉

_{Reviewed by React Doctor for commit 5d4856c.}

[posthog]

✅ Visual changes approved by @k11kirky — baseline updated in 5d4856c.

View this run in PostHog

2 removed.

[greptile-apps]

_{Reviews (1): Last reviewed commit: "Remove quick prompt header menu" | Re-trigger Greptile}

[k11kirky]

/trunk merge