redesign search trigger as wide input pill

[Flotapponnier]

Summary

Follow-up to #677 — the pill-redesign + lint fix landed after the merge so they need a new PR.

• Search trigger now reads as a real input: soft surface, rule border, max-width 440px, takes a flex-1 zone between nav and utilities (CMC-style).
• Platform shortcut badge (⌘K / Ctrl K) removed. `useSyncExternalStore` machinery gone with it.
• Side fix: `useEffect(() => setOpen(false), [pathname])` in site-header.tsx removed. It triggered the `react-hooks/set-state-in-effect` rule that's now failing CI on dev (PR fix(template) + header redesign (CMC-style + sub-nav) #677 merged despite the red lint). Mobile nav items already call `setOpen(false)` in their own onClick, so closing on route change works without the effect.

Test plan

• Visit any page on the Preview URL → search trigger reads as a proper input (light grey background, border, placeholder "Search benchmarks, products…").
• No `⌘K` / `Ctrl K` badge.
• `pnpm lint` passes (was failing on dev).
• Cmd+K / Ctrl+K shortcut still opens the dialog.
• Mobile menu closes on link tap.