Is there research that shows that a well maintained API adds value to large established companies?

Maintaining a good API for an ever changing product is VERY non-trivial. Designing easy to consume APIs and writing good documentation requires some specialized skills. Internal APIs are often very different than external ones. Some times APIs need to be duplicated which can make breaking changes of the underlying systems more complex.

I can see the value of APIs for a growing company that is trying to establish itself, but there must be a line somewhere.

I suspect a large amount of API support over the last decade has been driven by brand development and recruiting.