You dont "force your site to pay these costs", you design your clients as consumers from day one and get consumable external APIs for free.

This is a common "holy grail" that I've never seen any company pull off successfully. Like others have said you rarely have the ability to build a simple-yet-powerful API that perfectly services both your company proprietary apps and 3rd party clients. The requirements are almost always at odds with each other.

It certainly takes a huge amount of discipline, which is wholly absent in most service companies.

Hiding the costs, minimising the costs or "integrating them well with your platform" do not erase the costs.

Do most companies need an external API?

3/4 places I've worked have an external API. It might not be "public" but often enterprise customers want to pull their data for their own mangling.