Reusable rockets were "invented" by Lars Blackmore when he was working at JPL (Jet Propulsion Lab). I say invented because like anything in the evolution of engineering, credit is messy.

https://en.wikipedia.org/wiki/Jet_Propulsion_Laboratory

> Founded in 1936 by California Institute of Technology (Caltech) researchers, the laboratory is now owned and sponsored by NASA and administered and managed by Caltech.

Minimum-Landing-Error Powered-Descent Guidance for Mars Landing Using Convex Optimization http://larsjamesblackmore.com/BlackmoreEtAlJGCD10.pdf

Elon originally wanted parachutes and was convinced by Lars to go with self landing rockets.

Cheap reusable rockets allow for a lot more research for a lot less money.

Unfortunately, as the early history of SpaceX shows, it required a lot of failures to learn from to design the current crop of rockets. And that's the advantage that private R&D has... as long as the person in charge has money, failure is an option, because in anything publicly funded, any failure will relentlessly be blamed on the currently governing party by the opposition.