Hey,
The problem with offering everything under the sun would actually would be detrimental to the company in question, as there are three variant core competencies/operational pillars at play here in your ask.
Core Competency 1 - Manufacturing (Printing, Shipping/Logistics, Warehousing, Sourcing etc.)
Core Competency 2 - Information Technology (Website, API, Servers, Support, Updates etc.)
Core Competency 3 - Payment Processing (which is a completely different ballgame)
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Now what you're looking for can easily be achieved. Because your first point (having an API) solves most of your other points.
Just adding another specialist to the mix, solves your problem for you.
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Core Competency 1 - Manufacturing
For - Manufacturing - the behemoths in the market are Printful & Printify.
Both of these companies offer a sturdy API that helps you connect to Shopify or WooCommerce or even Amazon marketplace.
Core Competency 2 - Information Technology
For - Information Technology - there are two paths you can take.
First being the obvious - Shopify. You'll be able to get up and running in no time. Both the above have apps on the marketplace that do all the heavy lifting for you.
Printful -
https://apps.shopify.com/printful
Printify -
https://apps.shopify.com/printify
Second, you can go for Woocommerce - in a self hosted WordPress environment. Once again, you have plugins to manage everything.
Printful -
https://wordpress.org/plugins/printful-shipping-for-woocommerce/
Printify -
https://wordpress.org/plugins/printify-for-woocommerce/
There are several other directions you can take too. Here is a full list of integrations
Printful -
https://www.printful.com/integrations
Printify -
https://printify.com/integrations/
Beyong that, there's raw API availability for both providers
Printful -
https://www.printful.com/api
Printify -
https://printify.com/printify-api/
Core Competency 3 - Payment Processing
If you use Shopify - it has Shopify Payments - which is basically just Stripe wearing a Shopify T-shirt. Then there's PayPal and so many other regional players.
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Core Competency is a very important aspect of business. You've got to do something and become the market leader in that space. As against, running sort of a consulting business that "says yes to everything".
Printful is an excellent manufacturer. It hired a small team to develop a system that plugins into most marketplaces in the world. So while it became IT Adjacent, it is not an IT company.
Competing against Shopify, by setting up their own e-commerce system would require them to deal with a plethora of compliances, headaches, support requests, resources etc.
As for Payment processing, even Shopify couldn't dare go against someone like Stripe (even though they easily could). Instead they partnered with them. So while Stripe makes all the revenue, does all the due diligence, handles all the compliances - Shopify ends up getting a cut of the lot.