feat(shell): make search a button and a modal where searching applies

The phone's top bar is about to go, and the search box is the one thing
in it that is an action rather than chrome. It becomes a button in the
row that already says which page you are on, opening a wa-dialog with
the real search box in it.

Three decisions worth the words.

**A wa-dialog, and that is a mechanism rather than a taste.** wa-popup
renders `<div popover="manual">` and feature-detects the Popover API,
falling back to `strategy: "fixed"` where there is none -- which is
Chrome 113, the reference device, since `popover` is Chrome 114. And
`position: fixed` escapes ancestor overflow but not `contain: paint`,
which `.main-panel` carries, so a popup-shaped search panel opened from
a view's header is structurally clipped on that device. `<dialog>` /
`showModal()` is Chrome 37 and uses the real top layer. No tier here can
see the difference -- CI's Chromium and WebKit both have the Popover
API -- so the component test asserts the *mechanism*, a native
`<dialog>` in the tree, rather than the symptom.

**An element, not a PageAction.** Two of the seven searchable views are
detail views with no page-header; they filter on the term and say so in
their own headers. Declaring search as an action would mean seven hosts
each writing it out, which is a second list of searchable views, and it
would put a phone mode for actions inside page-header, which that
component documents its refusal to grow. search-store's own map is the
condition, asked by one component placed three times.

**The modal carries the real search-bar**, so there is still one
debounce, one clear button and one view-scoped placeholder. Escape
closes it and *keeps* the term -- the input treats Escape as "clear the
search", which is right in a header where the box stays on screen and
wrong in a surface whose dismissal would then discard the search.

frontend/src/components/page-header/page-header.ts

@@ -11,6 +11,7 @@ import {
     contextMenuStyles,
 } from '../../utils/context-menu-controller';
 import { ICON_MORE_ACTIONS } from '../../utils/icon-language';
+import '../search-dialog/search-trigger';
 
 /**
  * The one arrangement every primary view uses to say what it is.
@@ -457,6 +458,20 @@ export class PageHeader extends LitElement {
                 ${this.renderCount()}
                 <div class="spacer"></div>
                 ${this.renderScope()} ${this.renderSort()}
+                <!-- #57. Below 600px the top bar is out of the layout,
+                     so the search box has to be reachable from here.
+                     It renders nothing at every other width and on
+                     every view search-store says has nothing to
+                     search, which is why no host declares it: the map
+                     of searchable views already exists and this is one
+                     more reader of it, not a second copy.
+
+                     Before the actions, and never one of them: an
+                     action can collapse into the overflow menu, and on
+                     a phone that menu is already where the page's own
+                     actions live -- search behind an ellipsis is the
+                     top bar's problem moved rather than fixed. -->
+                <search-trigger></search-trigger>
                 ${this.renderActions()}
                 <slot name="actions"></slot>
             </header>