invite_stream_picker: Don't rely on custom code in input_pill.

By using a custom `generate_pill_html`, we can remove the
`has_stream` logic in `input_pill`, as part of a wider effort
to remove custom pill logic in `input_pill`.

web/src/invite_stream_picker_pill.ts

@@ -1,3 +1,5 @@
+import render_input_pill from "../templates/input_pill.hbs";
+
 import * as input_pill from "./input_pill";
 import {set_up_stream} from "./pill_typeahead";
 import * as stream_data from "./stream_data";
@@ -49,6 +51,12 @@ export function create($stream_pill_container: JQuery): stream_pill.StreamPillWi
         $container: $stream_pill_container,
         create_item_from_text: create_item_from_stream_name,
         get_text_from_item: stream_pill.get_stream_name_from_item,
+        generate_pill_html: (item) =>
+            render_input_pill({
+                ...item,
+                has_stream: true,
+                display_value: stream_pill.get_display_string_from_item(item),
+            }),
     });
     add_default_stream_pills(pill_widget);
     set_up_pill_typeahead({pill_widget, $pill_container: $stream_pill_container});