The infant industry argument always seemed odd to me. If the investment makes sense why wouldn't it happen anyway?
High-value added industries require enormous sums of investment which need to be sustained over long periods of time. Without infant industry protection, it's hard for me to understand how this kind of investment could be perceived as viable or worthwhile. By "high-value added industries", I don't mean plants that assemble parts or plants that produce individual parts but rather an entire supply chain contained within a domestic economy.
If it's not viable or worthwhile, then why do it?
Because it will become viable long term. South Korea did it very successfully.
^^^
Nearly developed nation that produces and exports "high-value added" goods protected infant industries from foreign competition in the past. Although I haven't looked at the economic literature on this subject, I think that economic theory could support "infant industry protection" as a viable economic strategy.
Why doesn't anyone want to get this long term profit then?
For one, the long term profit couldn't be realized without infant industry protection: assuming that there would be no tariffs or subsidies, there'd be no ability for a relatively new firm to compete against a relatively old firm for a variety of reasons. Also, investors are not willing to make this kind of commitment due to the long time horizon for the maturation of the investment. Investors would rather seek quicker short-term profits elsewhere for simple behavioral reasons: investors have a limited lifespan.
I apologize for my "on a napkin" argument but I think this makes intuitive sense, even within the confines of neo-classical models.
It might yes, but it isn't intuitively super convincing for me. People also have a limited life-span, after all. It isn't obvious that the current generation in the country should forego benefits for the sake of some far-off future generation.
Companies do long-term investment fairly often after all. And shareholders might hold their shares longer than a politician stays in government. 
1) Protecting infant industry is a strategy normally pursued by nationalist dictatorships with their eyes set on long term goals of creating a strong and broad based national industry. But the gain would generally materialize within 1-2 generations, so the "do it for your children" argument would come into effect.
2) Companies can not make long term investments if they go bankrupt in the meantime. Which they would if they tried to establish themselves in an industry with well established players, like the Koreans did in shipbuilding. If you want to become competitive in an industry with well established players with large resources (with the long term benefits that entails), there needs to be some kind of protection in the start-up phase.
