Nominal GDP doesn't matter much, but the nominal size of the market matters a lot for international power. Many countries want access to the American market because they can sell a lot of stuff which is converted at nominal exchange rates into their home currency, not at the theoretical PPP exchange rates. As long as China keeps the RMB undervalued, the Chinese market remains less attractive for international exporters than the American one. China's imports have almost caught up with American ones, but if the RMB was allowed to rise just a little, those imports will rise above American imports, making China the most attractive market for other countries.
Obviously, if you're looking at the economy in terms of the ability to fight an industrial war, you should really look at industrial production and volumes of production rather than price weighted GDP, whether it's PPP or nominal. And if you're thinking about wellbeing, PPP per capita is probably best and yes, developing countries tend to have a much larger informal sector, which is not counted in official GDP numbers
Obviously, if you're looking at the economy in terms of the ability to fight an industrial war, you should really look at industrial production and volumes of production rather than price weighted GDP, whether it's PPP or nominal. And if you're thinking about wellbeing, PPP per capita is probably best and yes, developing countries tend to have a much larger informal sector, which is not counted in official GDP numbers