> It looks like having a higher HDI simply allows countries to contribute more money to research - which some of them choose to do and others don't.

It could be even more complicated than that. R&D could lead to higher HDI, but there could be structural problems that prevent both of them from increasing. Governments in now HDI countries may struggle with state capacity to collect taxes/revenue and not be able to subsidize research and are also not able to provide basic rule of law and order for development. There may be problems with corruption and private R&D might not have any returns because other politically connected businesses will steal their innovations with impunity.

I'd be willing to believe that R&D/investments in technology can cause higher development. You can look at the late developing east Asian countries (Japan, South Korea, Singapore, Taiwan) as a possible example. The problem is that most poorer countries have problems that prevent development or sustained investment in R&D.