You really think NASA wasn't bureaucratic back in its glory days?
There were two big differences between then and now: then NASA had big-time funding and a clear mission that was stable over the course of a decade.
Bureaucracy is the optimal design pattern for an organization that has to accomplish a fixed set of objectives within a predictable set of constraints and with a stable set of resources. When bureaucracy works well, it becomes practically invisible. For example one of the greatest innovations Scotland Yard introduced to crime fighting is record keeping. That enables them to correlate crimes with evidence from past crimes in their records, including fingerprints. Before Scotland Yard introduced filing to police work, cops didn't fill out reports, and if a criminal got away with a crime the detectival slate was wiped clean for his next crime. But we usually don't think of a well-functioning police force as a record-keeping organization. When it works, you don't notice it.
Bureaucracies under stress are very noticeable. That's why the military is notoriously bureaucratic, because the bureaucracy's need to make it's work predictable and repeatable conflicts with the fluid nature of war. So "bureaucracy" in our minds is associated with malfunctioning or even corrupt organizations.
Bureaucracies are predictable. If you underfund a bureaucracy, it will inevitably put more effort into preserving itself and less into what it's supposed to do. If you ask it to accomplish something the people working in it perceive as impossible, false or contrived information will inevitably clog its record keeping system. If you demand schoolteachers raise score levels, score levels will rise whether or not children are getting better educated. If you demand cops increase the numbers of arrests, the numbers of arrests reported will rise whether or not there is an impact on crime.
So what happened to NASA's bureaucracy? Well, in the glory days, the environment was optimal for the functioning of bureaucracy. Even disasters like Apollo 1 weren't "unpredictable" from a planning standpoint. Things like that were bound to happen. The instant one did, the bureaucracy leapt into action, forming committees and tasks forces, like white blood cells rushing to the location of a wound. But once we'd landed on the Moon, things changed. Funding dropped, and the bureaucracy was asked to deal with conflicting priorities. Conflicting priorities aren't one of the things bureaucracies can handle.
You can't ask a bureaucracy to fix itself. Fixes have to come from a higher level. A company with a malfunctioning management structure needs to be addressed by the board. A malfunctioning government agency has to be addressed at the political level. It's not bureaucracy per se that brought low, it's politicians who don't want to make tough decisions about NASA priorities, nor adequately fund all the agency's objectives.