Console polish found while verifying Task 6688

Found while visually checking Task 6688 on a local stack. Itemized suggestions: 1. Directory list pane (Projects/People… strip) at 1024px with a detail pane open clips the second tab to "Peo" behind a scroll chevron; wrap it (the shared `Tabs` `wrap` prop) as the project and org detail strips do. 2. The console logs an Emotion warning on every render: `The pseudo class ":first-child" is potentially unsafe…`; switch that selector to `:first-of-type`. 3. Opening a project in the operator org logs a 403 for `GET /api/projects/{id}/live-cast` on every visit; ask only when the seat can see Live Cast, so the console stays clean. 4. Plan editor (Products › Plans › Edit): the Compliance and Node types groups still stack one switch per row while Features uses a two-column grid; lay them out the same way. Out of scope: plan switch behaviour; Project memory. Done means: - Each item fixed with a test, or declined in the PR body with a reason. - `./dc check full` green; visual check of each touched surface at desktop and 390px.

PR #5636 merged. Fixed Directory tab wrap, Live Cast 403 (live_cast_offered) and plan switch grids; :first-child was already fixed on develop. Local full gate timed out, no failures; CI green. Visual check clean at 1024/390. Deploy superseded. Follow-up draft #6894.