In order for the shipping rates to appear on the calendar, you must ensure that whatever carrier service you have integrated with your store returns a "delivery_date" attribute in the shipping rate response.
This is standard among all major carriers native integrations with Shopify, such as UPS, USPS, Canada Post, DHL and Fedex, but flat rate and weight based shipping rules do not return this data by default.
Similarly, there are many apps available that are able to provide customized shipping rates to your store. Some of these apps are able to be configured to return a "delivery_date" in the response, which can allow you to have flat rate shipping rules that return a delivery date response.
If you are configuring custom shipping rates through an app, this resource will help in determining how to add the delivery date to the response: https://help.shopify.com/en/api/reference/shipping-and-fulfillment/carrierservice