I believe this to be a bug, but have a work around you can try out that worked for me. Here's the steps to success:

1. Drop WebScheduleInfo onto the Design view.
2. Drop WebScheduleSqlClientProvider next to the WebScheduleInfo.
3. DO NOT SET a DataSourceID for the WebSceduleSqlClientProvider, but make sure you have the WebScheduleInfoID referencing the WebScheduleInfo you dropped in the first step. (Note: For some reason if you reference a connection string with WebScheduleSqlClientProvider when your calendar is being used on a content page of a master page it will *always* fail to connect to the datasource).
4. Drop a calendar or any of the schedule controls into the design page view. Make sure its WebScheduleInfoID is set to the WebScheduleInfo from the first step.
5. Right click in design view and click on View Code. Paste the code below into your Page_Load event.

WebScheduleSqlClientProvider1.Connection.ConnectionString = ConfigurationManager.ConnectionStrings["SweetSuccessConnectionString"].ConnectionString;

Be sure to change the connection string name to whatever it's called inside your Web.Config file. Now view the page in the browser and watch the magic happen!

Your code shows one way of setting the Connection.ConnectionString on the WebScheduleSqlClientProvider.

Two other ways the data provider can be told how to connect to the database are by setting the DataSourceID (which the data provider uses to retrieve the Connection String, however as described this will not work unless the fully-qualified ID of the Data Source control is used), or passing the connection string at run-time by the Connect method.

If you do not tell the data provider how to connect by one of the available means, it can't connect.