I cannot customize the checkout page with date or time selection

Supertime works as a cart-based app. Unfortunately, since Shopify requires Shopify Plus access for modifying the checkout page, we do not support this currently.

If you are using Supertime on different channels other than your online store, Supertime will only collect dates if your channel is showing your store's cart page.

If you’re looking to make sure your customer is able to see their selected time slot, this article will help you with showing your Supertime details in the post-checkout confirmation page:
Add Supertime details to the post-checkout confirmation page