The ReadingLists extension currently exposes an action API module implemented (see ReadingLists), which exposed through a REST API in RESTbase which also adds summaries to the output (see https://github.com/wikimedia/restbase/blob/master/v1/lists.js#L159 lists.js).
To unblock the sunsetting of RESTBase, the ReadingLists extension should implement its REST API in PHP and expose it through rest.php. This should be relatively easy by sharing code with the action API modules that are already implementing the logic.
The public endpoint URLs should be kept intact by applying the appropriate routing/rewrites in ATS or an API gateway. The new endpoint should follow the spec of the existing REST endpoint, see https://github.com/wikimedia/restbase/blob/master/v1/lists.yaml and https://en.wikipedia.org/api/rest_v1/#/Reading%20lists