Why are the local delivery and local pickup options missing from checkout when the customer decides to create an auto-charging subscription?

Not having the local delivery and local pickup option for auto-charging subscriptions is one of the limitations of the Shopify Subscriptions. You can see the limitation listed here: https://help.shopify.com/en/manual/products/subscriptions/setup#eligibility-requirements

If you currently use only these two local delivery options and are looking for a solution to allow your customers to checkout with auto-charging subscription products, then the recommended workaround for this is to create a standard shipping rate in your shop, so your customers will be able to complete the checkout with auto-charging subscription products.
But a thing to note here is that for auto-charging subscription products, Shopify always shows only the cheapest shipping rate in the checkout.