In terms of an alternative elder care solution, unfortunately, the only solution would be either to improve the health of the elderly so that they are able to keep working until they die, or continue as we are continuing to put a further strain on younger generations until the system basically collapses under the strain.
In regards to the second point, about international borders, once developing countries, which have rising populations, reach developed status, they will encounter the same problem that developed countries have i.e falling populations. So, to take advantage of countries with rising populations would simply be plastering over the birth rate problem in the developed world and thus pushing the problem into the future — never a wise tactic.
Not just that, but as mentioned in reply to Nell’s comment, the problem with immigration as a solution is it takes skilled workers and motivated workers from developing countries. This hinders developing countries from reaching developed status because it takes their best workers from them. As you say, the world is overpopulated, and the population will continue to grow until all nations reach developed status. So, immigration as a solution merely pushes the problem into the future while ensuring that the world continues to become further overpopulated by holding back developing countries from reaching developed status.
In regards to your final statement, the goal is not to push women into pregnancy, it is to create a world where those who wish to have children are able to have them, it is about creating a world where having children does not push people into poverty. After all, the human race without a sustainable birthrate is basically on the path to extinction. Or rather, the human race without babies is extinct.
So, indeed we do need to work on solutions, and the thing we need to find a solution for is how to make having children once again affordable for everyone so that couples feel able to have children again, and in enough numbers to sustain the world we have created. For this to happen, women need to average 2.1 children each. If we are to create a model that results in couples attaining that number, we have a lot of work to do to make technology start bringing the cost of living down rather than constantly pushing it up.