We're currently building a Private app (using Storefront API) for our self-deployed "platform" to support alternative way of selling products from Shopify without actually going to Shopify official Storefront. We're doing just fine and we're building this for our pilot customer, but next ones pile up.
It is ok for us to manage stores of our customers (and create Private apps for them), but currently we're using checkout.webUrl to finalize orders once cart was completed.
Now Shopify documentation is not clear to me, on how can we switch to our own checkout process so instead of using webUrl – provide our own shipping and payment forms?
What requirements do we need to meet or how we'd must change our Private app implementation to make this happen? Do we need to become Public app? We don't need to be listed in Shopify Store. Custom app? Using what API? "Proxied" Admin API?