> All of your probes and colony ships have to reproduce the source culture - or some viable subservient spin-off - with 100% accuracy.

Any empire/political system/civilization needs some means of holding it together. Traditionally this is done through armies. As our transport technology has gotten better, our ability to move armies around the world has increased and with it the ability to control large economic systems without them breaking apart. Nowadays we have a globe-spanning economy. However, time scales of interstellar travel are so big that cultural change is much faster than the method to build a colony. By the time that nuclear bomb ship has arrived at the other planet, we might have changed our mind threefold.

In fact, already if you send a generational spaceship to a star 100 ly away to reach its destination in a thousand years, by the time you receive message from their arrival, both their as our language will have evolved by 1100 years. So yeah you are right, cultural decay (or rather "change") is a big problem.

Currently our governments are run by humans with all the human failings. Instability, corruption, etc are components of the governments due to our human nature. You are right that it would be hard for human run governments to colonize the galaxy as due to the light speed limit, we can't achieve transport scales needed for a human-run civilization.

However, we won't need humans to run the governments. In the future we'll be able to engineer and build extremely intelligent AI systems with multiple layers of error checking to withstand decay for trillions of years. You think that sounds undoable? Check how many cells humans are made out of and how well they work together for decades.

The unsolvable political problem will just be turned into a solvable engineering problem.

Those AI systems would run key parts of the governments and economies themselves (entirely without needing human help) and enforce a set of basic rules like preventing nuclear weapons, bioweapons, torture, etc alongside ensuring they are the most powerful factor in those places, i.e. are able to fend off any kind of attack. Otherwise they won't interfere with human matters and let humans govern themselves with the human imperfections.

Basically, they'd provide the "galactic federation layer" and ensure peace through their programming. Some factors of their programming could be changed by the human population, but others, like preventing nuclear weapons, couldn't. So even if the humans wanted, they wouldn't be able to wage catastrophic war with the neighbour planet, or at least specific types of war (allowing trade war but not allowing nuclear war). Even if the risk for interplanetary war is just 0.1% per year, over millennia that's a large number and interplanetary war can have major consequences.

To the present reader, those AI systems probably seem weird and dystopic, but I think we are already evolving towards them. Already, many parts of the governments and economies of various countries are being automated. One day in the future the automation will have minimized human involvement enough so that you can think about von Neumann probes.

The first people to develop AI ran governments will probably be the dictatorships who want absolute control and micromanagement (single-operator government). But in general, the systems are more efficient than chaotic systems, and with increasing perfection of the technology, once a system is a dictatorship, it probably won't change back to a non-dictatorship without outside interference. It's important to eventually remove human control from them as otherwise over the millions of years, eventually someone will figure out how to flip the switch and create dictatorships. So it's kinda important to get the "right" set of rules into these computers.

Already building these computers is a huge project that would take a long time to get it right. But it's an easier task for humans to do than running an intergalactic civilization.

Also note that those systems don't have to necessarily deal with humans in their current form. They could also deal with human simulations uploaded to the cloud and the systems would just provide the resources to run that cloud. The "real world" might become utterly uninteresting to humans.