I want something that isn't focused on politics or the cold war more than necessary but is more of a technical and organizational account of how NASA and other agencies actually worked from the 1950s-1980s.
For this I highly recommend Digital Apollo, by David Mindell. There's some organizational history, but the technical story is that of the development of controls and systems engineering, when the task of flying a space mission became far too complex for manual guidance, the flow of data too huge for simple radio telemetry. It could only be handled by developing computers and software to do it. It's a commonplace that the space program had many valuable technological spinoffs, but this one has to be one of the most important and the least noticed.
The Outward Odyssey series has some really good books depending on what time periods you're interested in. Some of their books are a bit weak IMO (the one about the early history of the space shuttle was very disappointing) but others like Homesteading Space about Skylab are excellent.