feat(remote): the app/activity picker, and a protocol transport that really connects (t15)

The user's remote displayed but offered no app picker, and the protocol
transport was never actually switched on. Both halves are fixed here.

REMOTE PARITY. The entity declares RemoteEntityFeature.ACTIVITY — exactly
what core's androidtv_remote declares — and publishes activity_list, which
is what makes the frontend render an activity picker at all. The list is
the app directory (the user's apps map, else the curated table as a floor)
in map order, then the apps ADB discovery found and something can name, in
the one name chain (user map > device label > curated > package id).
Nothing nameless is listed as a bare package id, and current_activity is
None until something is known: no activity is invented. remote.turn_on
honours activity= by waking the device when it reports itself off and then
launching through the existing launch path; a listed name resolves
case-insensitively, a package id or a detail link passes through, and
anything else raises unknown_activity naming the value instead of silently
doing nothing. An empty/absent activity keeps the previous behaviour
exactly, async_turn_off is untouched, and a configured command script still
owns the sequence.

WHY THE TRANSPORT WAS DEAD, MEASURED. androidtvremote2's keep_reconnecting
only starts _async_reconnect, whose whole body is
`while self._remote_message_protocol:` — false on the first check with
nothing connected, so the task returned immediately: no connect attempt, no
retry, and the auth callback was never reachable. HA core connects FIRST and
only then keeps it alive. Our _start_protocol called keep_reconnecting after
generating a certificate and never called async_connect anywhere, so every
send_key_command/send_launch_app_command raised ConnectionClosed internally,
the helpers swallowed it into `return False` plus a debug line, and no push
(is_available/is_on/current_app/volume_info) ever arrived.

The fix is a background connect loop: async_connect under a bounded timeout,
network failures retried with the ADB watchdog's jitter and backoff,
InvalidAuth routed into the EXISTING reauth callback (entry.async_start_reauth),
and keep_reconnecting only after a session is up. Setup never blocks on a
transport. A send that cannot run is no longer silent: it is counted, its
error recorded verbatim, and logged at WARNING with the transport state, and
the remote publishes protocol_state / protocol_connected /
protocol_ever_authenticated / protocol_last_error / protocol_failures /
protocol_certificate plus sticky per-push *_seen flags — so "sent" and
"never switched on" can never look alike again.

CASE (b) IS FIXED, NOT DOCUMENTED: switching the transport on in the options
flow for an entry that never paired now routes into the real
async_step_protocol_pair, and "continue without pairing" persists the
transport OFF. One marker carries the pairing (CONF_PROTOCOL_PAIRED),
written by the setup pairing step, the reauth finisher and the reconfigure
finisher, so no entry can sit enabled-but-unpaired. The certificate from the
first pairing is reused by design, so a paired entry is asked for no new PIN
— which is also why the marker, not certificate material, is the source of
truth.

ADB STAYS THE PRIMARY STATE SOURCE. Arming the protocol introduces pushes
that were never live before, so the reading now names its tier:
manager.foreground_source drives the media player's app_source and the
remote's activity_source, and the new tests prove a full protocol push set
leaves the media state, the now-playing reading and every ADB store field
untouched, while an ADB-only entry still serves media player, sensors, apps,
scripts and receiver with only the protocol surface degraded.

Copy: unknown_activity, activity_launch_failed and protocol_unavailable in
BOTH strings.json and translations/en.json, pinned by a test; the README
documents the picker, activity resolution, how to read the transport-state
attributes, the script bypass (a script-routed press is never evidence of
pairing), the connect-then-keep-alive lifecycle and cert reuse versus the
other integration's own pairing.

Evidence: 6 new real-HA FlowManager tests in tests/test_config_flow_real_ha.py
(unpaired -> pairs, PIN -> marker, skip -> off, paired -> not asked again, and
the setup flow's marker on success and on skip), a real-HA entity contract
test in tests/test_ha_api_drift.py (exact supported_features equality with
core, activity_list/current_activity reaching the state machine, live _attr_
updates) and new suites in tests/test_protocol_transport.py (connect-first,
keep_reconnecting alone is a no-op, retry, auth rejection, honest send
failures, ADB-primary precedence) and tests/test_command_routing.py (the
activity contract, turn_on(activity=), script ownership, the transport
attribute states). No device and no browser exist here: device behaviour is
inferred from the library and core source, and every UI-facing claim is
proven at the entity/registry/metadata level under real Home Assistant.

Verify: suites 719/4 and 720/3; real-HA glob (tests/test_adb_command_real_ha.py
+ tests/test_config_flow_real_ha.py) 56 passed, 0 skipped; ruff clean; drift
exit 0 (25 modules, 76 imports, 206 relative imports, 1322 statement blocks,
57 call arguments) with the floors raised 40->70, 100->195, 500->1300 and
52->55.

tests/entity_stublib.py

@@ -243,7 +243,40 @@ class MediaPlayerEntity:
         return None
 
 
+class RemoteEntityFeature(enum.IntFlag):
+    """homeassistant.components.remote.RemoteEntityFeature (the real values)."""
+
+    LEARN_COMMAND = 1
+    DELETE_COMMAND = 2
+    ACTIVITY = 4
+
+
 class RemoteEntity:
+    """The RemoteEntity surface HA reads, backed by the ``_attr_*`` convention.
+
+    t15: HA renders the app/activity picker off ``supported_features`` and reads
+    ``activity_list``/``current_activity``; both are cached properties on the real
+    base, reading ``_attr_*``. The stub carries that contract (the same shape as the
+    ImageEntity face above) so a test can read what the entity publishes instead of
+    asserting against our own attribute names.
+    """
+
+    _attr_activity_list: list | None = None
+    _attr_current_activity: str | None = None
+    _attr_supported_features = RemoteEntityFeature(0)
+
+    @property
+    def supported_features(self):
+        return self._attr_supported_features
+
+    @property
+    def activity_list(self):
+        return self._attr_activity_list
+
+    @property
+    def current_activity(self):
+        return self._attr_current_activity
+
     def async_write_ha_state(self) -> None:
         return None
 
@@ -474,6 +507,14 @@ def install() -> None:
 
     remote = types.ModuleType("homeassistant.components.remote")
     remote.RemoteEntity = RemoteEntity
+    # t15: the feature flag that makes HA offer the activity/app picker, the service
+    # field the picker submits, and the two attributes HA reads off the base class.
+    remote.RemoteEntityFeature = RemoteEntityFeature
+    remote.ATTR_ACTIVITY = "activity"
+    remote.RemoteEntityStateAttribute = types.SimpleNamespace(
+        ACTIVITY_LIST="activity_list",
+        CURRENT_ACTIVITY="current_activity",
+    )
     sys.modules["homeassistant.components.remote"] = remote
 
     button = types.ModuleType("homeassistant.components.button")