Search widget: input claims role=combobox without required ARIA, and icon buttons have no accessible name

[Aureliolo]

The search widget renders two accessibility defects that axe-core reports. Both
are inside the widget's shadow root, so a site cannot fix them in a template.

Found with zensical==0.0.53 (modern theme), audited by Lighthouse 12.6.1 with
axe-core 4.13.0.

1. The search input claims role="combobox" without the required attributes

The input is constructed with role="combobox", but no aria-expanded and no
aria-controls:

N("input", { ..., placeholder: "Search", spellcheck: false, role: "combobox", dir: "auto" })

axe-core reports aria-required-attr:

Required ARIA attributes not present: aria-expanded, aria-controls

There is also no role="listbox" on the results container, no role="option"
on the results, and no aria-activedescendant, so the combobox contract is not
implemented anywhere in the widget. A screen reader announces a combobox whose
advertised expanded state never arrives.

The widget does implement arrow-key navigation over results, so completing the
contract looks like the right fix rather than dropping the role: mark the
results container as a listbox, mark each result as an option, and keep
aria-expanded / aria-activedescendant in sync as the user types.

2. The icon-only buttons have no accessible name

Both buttons beside the input render an icon and nothing else:

• the lucide-search button
• the lucide-list-filter button

axe-core reports button-name:

Element does not have inner text that is visible to screen readers;
aria-label attribute does not exist or is empty

They are announced only as "button", so neither is distinguishable by a screen
reader.

Impact

On a docs site gated on a Lighthouse accessibility budget, these two audits cost
20 weighted points and can push a page below a passing threshold. More
importantly, the search box is not usable with a screen reader as it stands.

Reproduction

1. Build any site with the modern theme and open a page.
2. Open search.
3. Run axe-core (or Lighthouse's accessibility category) against the page.

aria-required-attr and button-name both fail, with the offending nodes
inside the widget's shadow root.