In other words, why do countries generally have messy and complicated research funding systems like we do in the United States, where there are a bazillion different independent agencies and mechanisms for funding research, each with its own peculiar mechanisms and application rules? Wouldn't it make more sense to have some sort of unified science system where all of the science people can go and ask to get their science things funded?
I enjoyed thinking about and exploring this question, and so I wish to share my answer with you here as well:
First, let us consider why there are many organizations that fund research, rather than a single research-funding organization. This is a matter of evolutionary organizational structure. In most countries, research has a non-trivial budget and applies to many different concerns of government. That means there has to be some (probably largely hierarchical) structure for organizing it. Now, let's consider two prototypical organizational structures for government-funded research. First, we might have a general research agency, which contains subdivisions addressing the research needs of various other governmental tasks:
Alternatively, each government department might have its own research agency:
Almost everywhere, we see organizations more like the second structure than the first---there might well be some countries in the world where research is so small or so controlled that is it organized in the first way, but if so, I am not aware of them. Why might that be?
Consider what happens if you are a leader in the department of agriculture, and you want to expand your agency's research work. Unless strong regulation prevents you from doing so, it's much easier to create or expand a research organization within the agriculture department than it is to get an independent research department to do it for you. A research sub-department within agriculture is also more likely to serve the peculiar needs, time scale, market structure, etc. as relates to agriculture. It's also easier and more rewarding to go to government leadership and fight to get resources for your own organization, where you can explain exactly how you plan to utilize them, than to fight to give them to somebody else.
Since both government structure and research needs evolve over time, we may thus expect research organizations to multiply, both across the government as a whole and also within individual sub-organizations. They are, in fact, occasionally reorganized and combined with the goal of making them simpler and more efficient to interact with, just as other government agencies are, but that will typically not reduce the number down to one, just to a smaller "many." Moreover, we've only discussed government funding, not industry funding or funding by foundations and NGOs, which all have their own separate needs and desires and further complicate the funding landscape.
Now, to the second aspect of the question: why is there no central database for applications? Sometimes there are, at least partially. For example, in the United States all government requests for proposals go through FedBizOpps. Most research solicitations can thus be found there (though not all, due to the diversity of mechanisms), along with requests for things like security guards for the US Embassy in Costa Rica. As you might guess, however, the sheer breadth means this often isn't a terribly efficient method of searching.
Likewise, every agency has different sorts of information it's looking for in research proposals. Again, taking the US as an example, the NSF really wants to know how its funds will support graduate student and postdoc education, since that's a key part of its mandate. AFRL, on the other hand, usually doesn't care much about supporting students, and has a mandate instead focusing on how its funds will affect current military concerns. As a result, a "universal" proposal would likely be quite cumbersome even if the bureaucracies were somehow reconciled.
Bottom line: "research" is too complex and pervasive a set of needs to readily stay contained within a single unified organization.