Hi everyone -
Yes it is correct that we have mutations and information on how to complete the checkout through the API - however this is reserved for Sales Channels (public apps) as noted above. If you are creating a headless website for personal use / private app, this does not qualify as a sales channel. There is a permission required to process a checkout in this way, which is only able to be added to a sales channel app. There are also use cases for a native mobile app utilizing the Mobile Buy SDKs to require this.
The only option right now for web is to redirect to the WebURL.