I'm glad you brought this up, as it is very interesting and intricate. The issues I have with the statement(s) you've made basically come down to the fact that it's not really that simple. My area of expertise is Sub-Saharan Africa, where there are more than enough examples of underdeveloped nations that have stalled in the past decades. There's a few sides to this, namely:

1) The notion of the nation-state was, in an African sense, completely alien to the continent. The European colonial powers drew up a bunch of borders at the Berlin Conference in 1885. These borders had very little to do with where the various kingdoms and ethnic groups were situated and all to do with extracting as many resources as possible. So you had millions of Africans being told they were now countrymen of some other group of Africans hundreds of miles away, with whom they had no bond or cultural heritage. This resulted in a lack of political and cultural legitimacy for the colonial ruler, creating what's known as a gatekeeper state. There simply was no internal infrastructure, as everything was focussed on resource extraction and taxes from imports/exports. Once these countries started gaining independence they were forced to basically continue what the colonisers had started. In terms of nation building, not much had happened (perhaps a sense of shared struggle). There were next to no borders that existed before the colonial ones so there wasn't really anything to fall back on either.

2) The terms for independence were unbelievably to the detriment of the African states. France implemented an African 'franc' in its colonies and one of their terms for independence was that their colonies put all of their foreign reserve into the French central bank. Over the years this has been negotiated down to 85%, with them being allowed to access 15% annually. The French are effectively picking up 500 billion dollars every year from their former colonies.

3) Most of the "resource rich" states have had their resources privatised by multinational corporations, either because these corporations were a colonial inheritance, or there were no rules to stop them. A lot of income has been lost this way. (Although Botswana is a positive exception to this and would be a really interesting case study for you!)

4) Generally speaking, state funded instutions won't result in rapid economic growth in underdeveloped nations. If there is no impetus to improve or no fear of failure then there will be no development. Expecting the state to be strong and make the correct decisions when the state actors themselves are not fully qualified is a recipe for disaster. Mismanagement has been an issue for decades.

5) Corruption is a big, big problem. The stakes are very high and there's a lot of money involved.

6) "Developed" countries' expectations of development or growth should not be projected on "non-developed" countries. They're not going to evolve over time to something that represents that of the "developed" countries. It's a dangerous path to go down.

7) Almost every country in the world is now taking part in the globalised capitalist economy. You have to wonder if the cards that have been dealt are fair, and if underdeveloped countries have a chance. This links in to dependency theory. Does the "West" prosper because the "rest" is kept dependent?

I know probably a lot of what I've written may not be entirely relevant but I've tried to paint a picture of why, in theory, yes you could be right, but also why these things are so unbelievably complicated sometimes that it just doesn't work.