As often with your questions, I can't tell whether you are making simplistic or outright incorrect assumptions for the sake of asking a potentially interesting question on a theoretical level or if you really believe all this but the reality is that nobody mounted a political argument for doing it one way rather than the other because it's not an alternative any country faces at the moment. Germany in particular hasn't been actively encouraging refugees to enter the country and it has been trying hard to offload applications to third countries (Dublin system, agreement with Turkey) but that's simply not working.
The country also supports the UNHCR and programs to help refugees abroad. You could argue that spending more money on the that would make sense because each euro you spend on foreign aid helps more people than domestic programs, at least in the short term. But this type of approach (which has also been tried to manage regular migration flows) typically fails to have a significant impact on the number of new arrivals. So whatever you do in this respect, you still have to somehow deal with the people who make it to Germany and/or the Schengen area.
Now, some of your comments suggest you are actually contemplating another idea, namely forcing these refugees/asylum seekers to go to third countries and fund refugee camps there. That's legally and practically unrealistic but above all, it wouldn't even necessarily cost less. That's because in this case, you also have to factor in the cost of the removal itself (legal proceedings, running detention centres, paying for flights, police escorts, etc.) According to the estimates I have seen, that alone costs several tens of thousands of euro per person. If I go by your own €10000 estimate, that's the equivalent of several years of support, by which time most refugees will be working and net contributors to the economy and/or welfare system.
Finally, one reason everything costs more in Germany is because Germany is a rich country. And the reason for that is that workers in Germany are much more productive than workers in Turkey or Jordan. So someone who moves to Germany and finds work there will produce much more than what they would have produced abroad, making the world as a whole richer. That ought to be discounted from the costs side of the equation.