I'm going to take a stab at this one, but fair warning it's a fascinating question without easy answers. I'll start by pointing out that your question contains an anachronism; the germ theory of disease, so elegantly described in Koch's postulates, is a creation of the late nineteenth mostly by Koch and Pasteur. To quote below:

1. The microorganism or other pathogen must be present in all cases of the disease

2. The pathogen can be isolated from the diseased host and grown in pure culture

3. The pathogen from the pure culture must cause the disease when inoculated into a healthy, susceptible laboratory animal

4. The pathogen must be reisolated from the new host and shown to be the same as the originally inoculated pathogen

When you mention a theory being proposed in the 16th century, I assume you're referring to Fracastoro and contagion. So Fracastoro was an Italian physician, and he was grappling with one of the biggest challenges to Galenic medicine, and that was epidemic disease. A very brief refresher, the medical world on the 16th century was still steeped in the tradition of Galen, and especially the four humors (I would actually argue that it wasn't until ACTUAL germ theory that these were finally killed off, and some strains still survive in psychology). So medical theory of the day stated that health was based on balances in the four constituent fluids of the human body (the humors). Disease happened when they were unbalanced, and various treatments sought to rebalance them (this is the justification behind bloodletting, for example). This could explain almost any disease -- though you can see how epidemics challenged this. What factor was causing a variety of people to become sick at once? The Galenic explanation was miasma -- bad air, essentially, that would cause many people to fall ill.

However, by the 16th century, physicians had made careful observations of many epidemics, especially syphilis, and it was increasingly clear that some epidemics were spread by physical contact. How could this fit into Galenic medicine and the theory of miasma? It's in the context that he published "On Contagion", in which we first get the word "germs". He purposefully based this on the idea of plant breeding. Essentially, he argued that there were chemical substances called germs, and each different disease had a different germ. This substances could travel between person to person and caused "putrefaction" in the individual, making more germs. Thus, Fracastoro was ADDING on to Galenic medicine and the humors to explain his observations. Contagion became quite popular, and lively contagion versus miasma debates continued well into the 19th century (again, they weren't really put to rest until Koch).

So contagion is invented in the 16th century. HOWEVER, I would say that it's a mistake to see contagion as a precursor to germ theory. Science isn't that tidy. A couple scientific trends converged in the late 18th century that inspired Pasteur and then Koch. The first is the increasing debate on abiogenesis, the idea that life could arise spontaneously. Experiments with maggots (Redi) showed that life could not come from nothing. This inspired Pasteur in particular that "fermentation" and "inflammation" in disease might be due to microorganisms, and his experiments showed that this was possible. The second is the increasing sophistication of microscopes, which allowed for a visible world smaller than the human eye can see. The third was particularly influential, and that's the idea of epidemiology -- using statistical analysis to study disease. The most influential here is John Snow, who used disease maps to demonstrate that cholera was transmitted through the water -- without any knowledge of microorganisms (a theory that neither fit into miasma or contagion, and was quite controversial for this). If you haven't read it, The Ghost Map is an AMAZING book about this.

All this is to say that Robert Koch needed ALL of this to start his famous experiments. He needed the idea of contagion (even though it was in some ways opposed to germ theory). He needed improved technology like microscopes. He needed Snow's ideas and epidemiology that suggested a variety of ways disease could spread. He needed knowledge of "animalcules" and the microscopic world to give a a mechanism for disease to spread. And probably most importantly, he needed Pasteur's experiments which proved that microorganisms were everywhere, and the mechanisms by which they spread. Were it not for all of this, he would never have done his experiments that identified anthrax, TB, and cholera. And were it not for those identifications, he would not have generated his postulates, which I listed above, that define germ theory, and fundamentally changed medicine.

PHEW. That was really long.

So the TL;DR is -- germ theory didn't exist in the 16th century. It didn't really exist until Koch and Loeffler in 1884. And it took "so long" because science is a process by which we try to understand the world around us, and it took a cascading series of experiments and observations for Koch to formulate germ theory. That doesn't mean that our predecessors were any dumber than us. Rather, it's an important lesson that "science" is not a compendium of knowledge, but rather a method -- a method that's still going on.

I hope that was helpful!

Also, if you're willing, could I use your question in my latest podcast? I'm recording today and it's about a similar idea -- how the humoral model of disease was still shaping medical thought up until the 19th century. Thanks!