Shopify Plus checkout will not let a theme hide Cash on Delivery. Payment and delivery customizations are Shopify Functions. Checkout Rules Pro stores the rules as JSON on a metafield and lets the Function read them at checkout. No theme code. No metafield, no operations.
Payment methods
Each rule names a method and an action. The Function lowercases the name and looks for a payment method whose name contains it. “Cash on Delivery” matches a hide rule for “cash on delivery.”
Hide removes that method. Reorder moves it to an index. Wallets are skipped on reorder: Shop Pay, Apple Pay, Google Pay, and PayPal. Shopify does not let those move, so the Function does not pretend.
The method has to be enabled in Shopify first. The app can only place or hide something checkout is already offering.
Shipping rates
Delivery rules use the same shape against rate titles. Hide drops the option. Rename replaces the title, and an empty new title is ignored. A carrier-calculated rate can still show the carrier in front of whatever you wrote. The Function changes the label Shopify gives it. It does not invent a new price.
I wrote the payment side up in how to reorder payment methods in a Shopify Plus checkout. The Function is a short loop over rules. The annoying part is matching names without matching the wrong method.