Fix widgets getting stuck in loading states (#31314)

* Refer to ClientWidgetApi as "widget API" rather than "messaging"

* Rename StopGapWidgetDriver to ElementWidgetDriver

* Rename StopGapWidget to WidgetMessaging

* Fix WidgetMessaging's lifetime by storing it in WidgetMessagingStore

(Rather than storing just the raw ClientWidgetApi objects.)

* Unfail test

* use an error

* cleanup start

* Add docs

* Prettier

* link to store

* remove a let

* More logging, split up loop

* Add a test demonstrating a regression in Call.start

* Restore Call.start to a single, robust event loop

* Fix test failure by resetting the messaging store

* Expand on the WidgetMessaging doc comment

* Add additional tests to buff up coverage

* Add a test for the sticker picker opening the IM.

* reduce copy paste

---------

Co-authored-by: Half-Shot <will@half-shot.uk>
Co-authored-by: Timo K <toger5@hotmail.de>

test/unit-tests/stores/room-list/algorithms/Algorithm-test.ts

@@ -12,7 +12,6 @@ import { KnownMembership } from "matrix-js-sdk/src/types";
 import { Widget } from "matrix-widget-api";
 
 import type { MatrixClient } from "matrix-js-sdk/src/matrix";
-import type { ClientWidgetApi } from "matrix-widget-api";
 import {
     stubClient,
     setupAsyncStoreWithClient,
@@ -29,6 +28,7 @@ import { Algorithm } from "../../../../../src/stores/room-list/algorithms/Algori
 import { CallStore } from "../../../../../src/stores/CallStore";
 import { WidgetMessagingStore } from "../../../../../src/stores/widgets/WidgetMessagingStore";
 import { ConnectionState } from "../../../../../src/models/Call";
+import { type WidgetMessaging } from "../../../../../src/stores/widgets/WidgetMessaging";
 
 describe("Algorithm", () => {
     useMockedCalls();
@@ -89,7 +89,7 @@ describe("Algorithm", () => {
         const widget = new Widget(call.widget);
         WidgetMessagingStore.instance.storeMessaging(widget, roomWithCall.roomId, {
             stop: () => {},
-        } as unknown as ClientWidgetApi);
+        } as unknown as WidgetMessaging);
 
         // End of setup