Why did RSA encryption become popular for key exchange?
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.