Summary

Fixes the AI assistant buttons not appearing in the Chrome Next header.

Relates to #260010

Problem

Chrome Next renders the header differently from the legacy chrome — it does not render HeaderNavControls (the chrome.navControls.registerRight mount points). All AI assistant buttons (Security, Observability, Search, AI experience picker) relied exclusively on registerRight, so they were invisible when Chrome Next was enabled.

Additionally, the existing chrome.next.aiButton.set() API was a single-slot, last-write-wins design — only the Agent Builder could use it, and other AI experiences had no way to register their buttons.

Changes

1. Multi-registration aiButton.register() API

Replaced chrome.next.aiButton.set(node) with chrome.next.aiButton.register(button):

• Multiple plugins can register buttons; each call returns an unregister callback
• Accepts ReactNode | MountPoint as content (via ChromeExtensionContent), so plugins can reuse their existing mount functions
• Internal state uses BehaviorSubject<ReadonlySet<ChromeNextAiButton>> to manage registrations
• Chrome Next header renders all registered buttons via AiButtonSlot

2. Dual registration in all AI plugins

Each AI plugin now registers with both chrome.navControls.registerRight (legacy) and chrome.next.aiButton.register (Chrome Next). This ensures buttons appear regardless of which chrome is active. All dual registrations are marked with // TODO: Chrome-Next hack comments linking to #260010 for cleanup once Chrome Next is the only chrome.

Plugins updated:

• elastic_assistant (Security AI Assistant)
• observability_ai_assistant_app (Observability AI Assistant)
• search_assistant (Search AI Assistant)
• ai_assistant_management/selection (AI experience picker)
• agent_builder (Agent Builder — already had Chrome Next registration, updated to new API)

3. Type consolidation

Moved all Chrome Next types into src/core/packages/chrome/browser/src/chrome_next/:

• ChromeNextAiButton, ChromeNextHeaderConfig, ChromeNextGlobalSearchConfig, ChromeNext
• Extracted InternalChromeNext interface in browser-internal-types

4. Empty mount point layout fix

Fixed phantom gaps in the Chrome Next trailing actions caused by AI buttons that register but render nothing (e.g., when a solution's assistant is not enabled). Applied mountPointContainerCss to both the MountPoint and ReactNode branches in HeaderExtension so empty wrappers collapse out of the flex layout.