There are browser-side APIs to generate key pairs, but there is no real standard as the KEYGEN tag is not widely supported, so on the server side you need to implement virtually one method per browser type you are addressing. And then you want to store your private keys somewhere safe, which is implemented differently by OS&#x2F;browser. Some browsers use the system keystore for storage, others like Firefox have their own implementation (NSS). We are not lacking implementations here, merely browser standards.
To answer your question: I do not think startssl supports local key generation for all OS&#x2F;browser combinations, they probably (painfully) hardcoded the most common ones.