Regarding your inquiry, the Delivery Customization API can be used for renaming, reordering, and hiding delivery options during checkout; however, it currently does not allow for adding descriptions directly.
Using the UI Extension, you can add a description below the shipping option through the UI Extension API:
Reference:
https://shopify.dev/docs/api/checkout-ui-extensions/2024-07/extension-targets-overviewCheckout Locations Shipping