The API is simply one of the important parts and success can be measured by how well it interacts with the other parts. Again, go back to the function of an API - an interface between the program and the computer.
Programmers write their code with a particular API in mind.
The API sets the rules and regulations of how things interact and who is in control. The success of an API can be viewed as who's rules are the simplest, the most well-understood, and maintains the best control over the entire process. If an API allows a program too much control of system resources and this causes O.S. failures, then that API is not doing its job properly and cannot be viewed as a success. If the API standards are not well understood, programmers will have difficulties writing programs that function properly. There are numerous examples of extremely popular products that are actually inferior to other existing products.