Migrate StartChat plugin to NativeInputStateProvider

[malmstein]

Task/Issue URL: https://app.asana.com/1/137249556945/project/1174433894299346/task/1214802413219916?focus=true

Stacked on #8556 — base is feature/david/native_input_tabid_non_null until that merges.

Description

First plugin migration onto the per-tab push model. StartChatViewModel drops DuckChatInputModeState and observes the tab's NativeInputState via NativeInputStateProvider.

• NativeInputState gets a stored toggleSelection: ToggleSelection field (defaults to today's context-derived value via NativeInputState.defaultToggleFor). toggleSelection is distinct from inputContext: the context says where the widget lives, the selection says which tab the user tapped — they can diverge inside a browser omnibar.
• NativeInputHost exposes val tabId: String. The widget implements it from activeTabId; reads pre-configure throw, mirroring getInputState().
• Widget's configureMainButtonsVisibility calls viewModel.setToggleSelection(...) on every onTabSelected / onTabReselected, before the existing updatePluginContainerVisibility call. WidgetConfig carries a nullable toggleSelection override; configure() resets it (null = follow context default).
• StartChatViewModel injects NativeInputStateProvider and exposes bindTabId(tabId); the combined isVisible flow flatMapLatests on stateForTab(tabId) and gates on toggleSelection == SEARCH.
• StartChatView stashes the host's tabId before attach and forwards it to the VM from onAttachedToWindow; the plugin's createView calls bindTabId(host.tabId) alongside the existing onIconClicked wiring.

DuckChatInputModeState stays in place — still consumed by ambient UI (NTP background logo). Removal happens once nothing reads it. updatePluginContainerVisibility carries a TODO: it becomes redundant once every plugin reads toggleSelection from NativeInputStateProvider and decides its own visibility.

Steps to test this PR

StartChat icon visibility

• Enable Duck.ai feature + user setting on; disable input-screen setting (search omnibar mode). On a website, focus the omnibar — StartChat icon visible.
• Same setup, tap the Duck.ai tab in the toggle — icon hides.
• Tap back to the Search tab — icon reappears.
• Disable Duck.ai user setting — icon hides regardless of tab.
• Enable input-screen setting (SEARCH_AND_DUCK_AI mode) — icon hides.

Toggle / tab switching across tabs

• Open a website on tab A, tap the Duck.ai toggle. Switch to a fresh tab B in the browser. The widget should reflect tab B's selection independently from tab A.

UI changes

No visual change to existing UI; refactor of how the StartChat icon's visibility is wired.

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Note

Medium Risk
Updates native input state modeling and plugin/widget wiring around per-tab state, which could regress toggle selection behavior and StartChat visibility across tabs. No auth, network, or data persistence changes are introduced beyond UI state propagation.

Overview
StartChat plugin visibility is migrated to the per-tab push model: StartChatViewModel now binds a tabId and observes NativeInputStateProvider.stateForTab(tabId) (dropping DuckChatInputModeState) to decide visibility based on inputMode and the selected toggle.

NativeInputState now includes a persisted toggleSelection (defaulted via defaultToggleFor), the widget ViewModel can explicitly setToggleSelection, and NativeInputModeWidget pushes the current selection on tab (re)select and exposes NativeInputHost.tabId so plugins can subscribe to the correct tab state. Tests are updated and a new StartChatViewModelTest is added to cover the new visibility rules.

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