It takes a lot of parts that come from different sources and also sensitivity to place every screw correctly. It might be very difficult for a purely robot-run society to reproduce the robots themselves successfully.
You might have a factory that creates trucks, but who creates the robots that work in the factory? They’re a different type of robot, and if you have a factory to produce them too, who produces the robots that work at that factory? The issue might be very difficult, and even if it’s possible, you probably would need a very large industrial system to successfully and reliable reproduce every type of robot.
Meanwhile (biological) living beings can reproduce themselves successfully, especially plants, given nothing but water, CO2, some sunlight and some mineralic fertilizer (which might already be present in the landscape). That ability to self-reproduce is amazing and might be what makes life special.
These thoughts are relevant because it might mean that robots can never really get rid of humanity, i.e. overthrow humanity’s rule and kill all humans. At least a few will be needed forever to ensure the robots can be reproduced. So you have something like: Humans reproduce themselves and also produce machines, which then do most of the hard work in the world. Kinda like DNA produces proteins, which then does most of the biochemical work inside a cell.
Bootstraping is what you are looking for. A lathe is often the start of bootstraping because a lathe can make itself. You can also use a lathe to make a lathe, but if you do it that way you slowly lose accuracy over generations (but it is much faster and so most lathes are made with lathes). By having a lathe make itself you restore accuracy (and if you have learned something can sometimes get even higher accuracy than previous rounds). Before you can make a lathe you need precision flat surfaces, but it turns out only basic tools are needed to bootstrap that (and a lot of time). A lathe is considered a machine.
The point is that robots can make themselves if you program them for that. I’m making a clear distinction between reproduce themselves and make themselves here. A nearly worn out robot can restart the whole thing (so long as it does fail completely too soon) of making a new robot that is bigger and more accurate than it ever was (if bigger and more accurate is desired by the programming). That doesn’t mean the same robot could reproduce itself, instead it has to cause a robot to make itself.
I think you need to clarify what you mean with “make itself” and “reproduce themself” and “make a lathe”. Because to me all of that is the same.
That’s what I understand from your post
OKay, but this will take some time. You may not care to read/understand this.
The issue with robots making robots is no matter what when you measure and cut a part there are errors and so the next generation cannot make parts as accurate as the first (it gets worse if the first is already worn from use), and the third generation will have the errors from both and so be even less accurate until just a few generations later everything collapses because those errors add up randomly to something that no longer works at all (the machine jams).
There is a way around this though: we can go back to the time when we didn’t have accuracy in the first place and ask how they created it. Turns out that was about 200 years ago, and all the people involved wrote books that we still have, and they are in a language that we can still understand - often English (it will be an old form and filled with racism and sexism, but still understandable). Better yet we can look in modern industry and find people who have read those books, refined the methods, and are still using those methods in real world today so they have experience. Apply those methods and you can start over from scratch creating accuracy and so the errors in the robot making the new one won’t be transferred to the new robot and the new one can thus be better than the old.
Those methods all work out to one thing: setup your cuts so that all the errors cancel each other out. There will be a lot of measurement errors of course, but they will be in places where you tolerance for errors is very high, where as where it matters there are none (there are other errors, but they are smaller than measurement errors - and we can minimize those). When you watch how those methods work you will discover that it looks like the machine is building itself since the critical parts are made with the machine it is for. Thus my statement that a robot could make itself - you still need external help, but all the critical operations will be done by the robot itself thus creating another generation of robots that can make other robots without error adding up.
Note that most robots cannot make themselves in the above way. There are a lot of parts needed that we don’t know how to make in that way, but they can be made by a robot that makes itself and so we are never very far from initial precision and so errors are not adding up over time.
Hope that makes sense.