You're reading more in there than I wrote. Europeans should not be grateful to the USA. A way of doing things is in use today, making the world a slightly better place for both Europe and the USA, maybe even for the rest of the world. Everybody might be grateful for that, but not towards any specific party. We all gain.
Some European countries have nukes anyway, I think it's officially limited to the UK and France. But Germany does not, which would have been unthinkable between world wars. Belgium, my own country, has access to Uranium via Congo, has all know-how as we (oopsie) did teach Iran the necessary tricks, has good enough industry to do the job, and has nuclear reactors. All the elements are there. In fact, it's 99% sure USA nukes are in the country, and if we really wanted, we should be technologically capable of stealing them and remaking them into a usable if less powerfull nuke. The only reason Belgium hasn't any nukes is because we don't want them and do our utmost best not to have them.
Now Belgium is one of the smallest and least powerful countries. EU and NATO live here in Brussels because most other countries would become too powerful. If we can build nukes, just about any other European country should be able to do the trick. Meanwhile, the only way the USA can keep Iran and North Korea from having the bomb is because the nations are badly functioning, and there aren't much of them. If 20+ reasonably rich and open European country decided to develop nukes at them same time, there is no way stopping all of them.
BUT If that happens, the very next war would be very very ugly, and we all know it. Part of the reason for the existence of the EU is giving us a way to project global power, but another part is that after WW2, European countries had to learn to live together, or we'd make the whole continent uninhabitable. We simply can't afford a WW3, and neither can the USA.