Government should not pick their people, but rather the people should pick their governments. If we went with this row of reasoning, governments would not be granted power to mandate who could live within their borders.
If you are not catching my point, let me be clearer. Governments would not be able to restrict people from moving in. Immigration would be open. Borders would be open.
There would be an exception. If we grant that governments exist to protect the people, then they should be allowed to screen out those who might do harm. If there were due reason to suppose the person coming would bring harm, then that country would have cause not to accept the individual.
That opens up the question of whether governments should be allowed to shield out immigrants for fear of overpopulation or fear of terrorists coming in. I will only say, I can see how those reasons can be bent into excuses. I can see how those fears can be latched onto and used to exclude people who will do no harm. If you exclude the whole, when it is but the one who will do harm, then you do harm, you exclude entire classes of people wrongly.
(Note: Blog rewritten morning of Jan. 28, 2017.)