feat: show a conversation its own background tasks, and give it back every outcome

An `execute_task mode="async"` was invisible from the chat that started it.
The only trace was the receipt in the transcript and a row on the Tasks page
— which does not say *which* of those rows the assistant just spawned — so
"is it still going?" had no answer where the question is asked.

Worse, a task that did not simply succeed never came back at all. `run_job`
branched on `Ok`/`Err` first and routed by `job.kind` only inside the `Ok`
arm, so a failure or a kill left through the `Err` arm's unconditional
`hub.notify` — the home source (`/sethome`), worded "Cron job … failed" —
while the parent conversation sat waiting for a `task_completed` that would
never arrive. The wrong chat, and a wedged one.

The fix is a shape, not a branch: one `JobOutcome` classification, then one
`match job.kind` delivery site for every ending. An async task now ends in
its parent conversation whatever happened to it. The sink has a single
channel deliberately — to the model reading it, "it broke" is a result like
any other and must not be overlookable — so a failure is delivered as prose,
carrying whatever partial output the run produced, which is usually the only
clue about why. A cron job keeps the home notification: it belongs to nobody's
conversation. Cancellation becomes a third outcome rather than a flavour of
failure (`job_runs.status` has always had `'cancelled'` in its CHECK and
nothing ever wrote it), classified off the new typed `TurnCancelled` error so
nothing keys on a message string.

The strip above the composer is the visible half. `ServerEvent::TaskUpdate`
announces state to the source of the parent conversation only; the list is
`renderTaskStrip` (shared by the desktop copilot and the mobile chat), fed by
state on `ChatSession`. Each row links to `#session/{id}` — the page that
already shows, live, what a background agent is doing, and without which
"a task is running" is a fact you can do nothing with. Stopping is the
existing kill endpoint. A finished row clears itself after 20 s (its result
is in the conversation by then); a failed one stays until dismissed, and the
dismissal is remembered across reloads.

`GET /api/{source}/tasks` is what makes the strip survive a browser refresh:
the event is a broadcast with no replay, so without a load-time read a reload
would empty a chat that still has work running under it. It answers with the
running tasks plus failures from the last 30 minutes — the two states a person
can still act on. Successes are absent on purpose. Its window compares through
`datetime()` on both sides: `completed_at` is RFC 3339 and the cutoff is
SQLite-shaped, and `'T' > ' '` would let every same-day row through a window
meant to exclude it.

Not addressed, and worth doing next: a cron job's result should go where its
creator says, not always to the home chat.

web/components/copilot.js

@@ -2,6 +2,7 @@ import { html, nothing } from 'lit';
 import { ChatSession }   from '../lib/chat-session.js';
 import { t, I18nMixin }  from '../lib/i18n.js';
 import { renderMsg, renderAttachmentChips } from './copilot-render.js';
+import { renderTaskStrip } from './shared/agent-tasks.js';
 
 // Built-in (server-handled) slash commands shown at the top of the composer
 // autocomplete. Custom commands (from `commands/<name>/`) are fetched from
@@ -59,6 +60,10 @@ export class AppCopilot extends I18nMixin(ChatSession) {
     this._onPageChange      = this._onPageChange.bind(this);
   }
 
+  // The desktop shell routes `#session/{id}`, so a background task's row links
+  // through to what it is doing.
+  get _canOpenTaskSession() { return true; }
+
   connectedCallback() {
     super.connectedCallback?.();
     this._restoreState();
@@ -398,6 +403,7 @@ export class AppCopilot extends I18nMixin(ChatSession) {
       </div>
 
       <div class="copilot-input-area">
+        ${renderTaskStrip(this)}
         ${this._renderNoModelsBanner()}
         <div class="copilot-composer"
              @dragover=${(e) => e.preventDefault()}