Aerospace Engineer here, I can give a bit of an answer to this from the technology/mathematics angle. There are many social reasons we didn't go to the moon until 1969 that I don't feel qualified to answer.

There are two things I would point to as the main reason:

• advancements in mathematics

• Advancements in Materials Science and manufacturing

So let's start with the math

First off we're gonna have to address that, on the surface, there isn't too much difference between a rocket/missile that is designed to stay close to the earth's surface and one that goes to the moon. The moon rocket is much larger though. But, in practice, you launch any rocket and it goes on a ballistic trajectory. Space rockets just kind of miss the ground.

So the "easy" part is getting your rocket up there. The hard part is figuring out how to do anything else. This is where orbital mechanics comes into play. Historically, around the same time as the Sibiu manuscript, work is being done on astronomy to better understand planetary motion. Johannes Kepler would famously publish his work describing this motion in 1605, 50 years after the manuscript is published.

Another important bit of math we need to discuss wouldn't come about (more correctly, be applied directly to if rockets can achieve space travel) would be the Tsiolkovsky Rocket Equation published in 1903 (British mathematician William Moore derived it earlier in 1810 for terrestrial applications). The equation is pretty simple equation shower it required math techniques, the natural logarithm, to derive which wouldn't be available until the 1700s via Leonard Euler

Now the rocket equation leads us into the second issue: What do you build your rocket out of?

Mass is king when it comes to space travel. The rocket equation I mentioned above relates the final velocity of a rocket, not to the mass of the rocket, but to the ratio of the initial mass:the final mass. I.e.: build your rocket as light as possible and fill it with as much fuel as you can. The equation also has a term for the initial velocity of the rocket. In practice this comes down to: you need a fuel that gives you a lot of power per kg of mass.

So to get reliable space travel you need a very lightweight rocket with a powerful fuel source. In the 1500s and earlier most rockets would be powered by black powder which... Is not great for space travel. For a comparison, a modern E class model rocket motor has a max thrust around 20 N. One F-1 from the Saturn V has a thrust of about 6.7x10^6 N (6.7 MN).

Additionally you have the need for an oxidizor. You can't just stick gunpowder (or Kerosene like the Saturn V first stage) into a rocket and light it, you need oxygen for it to burn and it wasn't until the early to mid 20th century with cryogenic storage advancements that have made this feasible.

And that's just the fuel! You need something strong but light to make the rocket itself. Iron and wood aren't gonna give you the weight or structural support you need. Both The US and Soviet Union used titanium for many critical parts of their rockets which in and of itself wasn't discovered until 1791 and the process of economicly extracting it in a usable form wasn't developed until 1910. Aluminum is one of the primary construction materials of modern rockets but again, it wasn't readily produced until the later half of the 1800s