These statements are indeed mythical. They may remain true, but that would be on us.
The first, that poor countries are destined to stay poor, has a complicated rub to it. If the resource and ecological situation looked more like the 17th-19th centuries, then no, it'd be simply grotesque if poor countries remained poor. However, now it may be that the whole planet must cut back on their living standards. It seems likely that poor countries would in any scenario be able to grow somewhat (while rich countries would need to make major sacrifices to their living standards), but it may not be possible for all of the world to live a European lifestyle.
The second again has a caveat: Foreign aid has to be done correctly. Many forms of foreign aid can lead to unnecessary projects instead of real infrastructure, the enrichment of tribal leaders and corrupt local and even national bureaucrats, dependency and the destruction of local industry.
The third is totally true. In fact, when people die, their family members tend to want to replace them. The best route to reducing population is to promote education, opportunities for women outside of the home, the growth of living standards, and contraceptives. When people have a choice, they tend to not have enough children to grow the population.