NASA didn't stop innovating per se. In the 1980s and 1990s they worked out their own version of vertical take-off and landing and actually test flew it:
https://www.youtube.com/watch?v=wv9n9Casp1o
Linear aerospike engines are pretty neat too:
https://www.youtube.com/watch?v=FcW9kUUTfxY
Problem is absolutely none of this stuff ever got close to operational. It kept getting cancelled in the prototype stage.
If the DC-X had been developed into a fully reusable first stage, we'd have had something like Falcon 9 by the year 2000 and might be on Mars by now.
The difference between SpaceX and NASA is mainly that SpaceX cares about cost.
Or we can say that NASA has incentives to absurdly increase cost, like any bureaucracy.
There is no way the DC-X could have been developed without increasing enormously the budget (and taxes from Americans). That was the reason it was not done.
And innovation when you have many, many billions is one thing, if you have to operate in a commercial market but still find a way to innovate at that level its quite amazing.
I have issues with how Musk communicates, but fundamentally I think Tesla is innovating as much as SpaceX. Many here and me have lots of issues in how Self-Driving is developed, but fundamentally its the right concept.
What Tesla is doing with batteries is quite amazing. I have over the last 2 years spent a lot of time understanding the battery industry and what Tesla is doing is actually incredibly impressive. People mostly don't understand that because battery and battery production is far more obscure as topic.
However whatever else you can say about Elon, he is committed to the projects that he is committed too.
SpaceX will not give up on Starship, or at most replace it with a changed design that tries to solve the same problem. Tesla will not stop trying to push down the price of battery, they are comited to it and are willing to go all the way to vertically integrate mining if that what it takes. Tesla will not stop developing Self-Driving, they will push forward and invest as much as they need to invest.
I do wish the Communication around the Self-Driving technology would change. They should have just called it 'Co-Pilot' and sold a 'Advanced Co-Pilot' and say they are working on 'Self-Driving' but it not a product (yet). Stop promising it will come so soon, and I not sure about how I feel about letting Beta testers put up videos.
In which way did he lead the program technically?