Currently? No, they've got no vessel that could do this.
In the future? Sure, they could, but it's kinda silly to use NASA's resources on shuttling cargo to/from low Earth orbit. That can and should be a routine thing that we offload to commercial entities, so NASA can focus on the risky, not-profitable, big-picture stuff.
NASA definitely moved fast for Apollo, but I wouldn't call it lean.
I think its biggest problem is that it's such a political animal. They can't just do things in the obvious and straightforward way. The work has to be spread around so that each congressman's district can get a piece of the pie.
> Maybe, it acquired too many leeches feeding from it to move fast and lean since the Moon landings times?
Yes, that's one part - but the "real" reason is risk aversity. Basically, as long as there was the UdSSR to beat in a space race, the American public was 100% behind NASA - no matter how much (money and lives) it'd cost. When the Soviets were beat square, there was nothing worth the "risk" so everything had to be done by the books, with certifications and all that other bureaucratic nonsense.
NASA's focus is not so much the engineering of repeatable launches. NASA's focus is the deep science of next-gen space travel and utility, solar system exploration, etc. There was a Reddit AMA recently where a NASA director said they're happy to help SpaceX innovate in rocket engineering and reducing launch cost, because that's not where NASA wants to focus.