Why did RSA encryption become popular for key exchange? This is a soft question.  I don't know a lot about cryptography or its history, but it seems like a common use for RSA is to do key exchange by encrypting a symmetric key to send a longer message (e.g., the description of iMessage here).  Isn't this exactly the thing that Diffie-Hellman key exchange, which is older (and to me seems simpler) is for?  Looking at Wikipedia, they were also both patented, so this wouldn't have been responsible for the choice.

To be clear, I'm not asking whether it's theoretically important that public key cryptography is possible.  I'm asking why it became a standard method in practice for doing key exchange.  (To a non-cryptographer, DH looks easier to implement, and also isn't tied to the details of the group used.)
There is no strong technical reason.  We could have used Diffie-Hellman (with appropriate signatures) just as well as RSA.

So why RSA?  As far as I can tell, non-technical historical reasons dominated.  RSA was patented and there was a company behind it, marketing and advocating for RSA.  Also, there were good libraries, and RSA was easy to understand and familiar to developers.  For these reasons, RSA was chosen, and once it was the popular choice, it stayed that way due to inertia.

These days, the main driver that has caused an increase of usage of Diffie-Hellman is the desire for perfect forward secrecy, something that is easy to achieve by using Diffie-Hellman but is slower with RSA.

Incidentally: It's Diffie-Hellman key exchange, not Diffie-Hellman secret sharing.  Secret sharing is something else entirely.