The EUV LLC research was US-funded, agreed. But the light source is Cymer, which ASML has owned since 2013. Its drive laser is Trumpf (Germany), and the optics are Zeiss (Germany). Nikon and Canon had access to the same public research and dropped EUV. The US had GCA, Perkin-Elmer and SVG, and lost the market anyway. If integration were the easy part, someone else would have done it. You are underselling what it means to perfect the machines by calling it "just integrating".
By that logic SpaceX is a German company. Liquid-fuel rocketry was researched and developed at Peenemünde, and the US brought in von Braun's team wholesale to build its program. SpaceX just integrates German IP. Or Google is Swiss. The web was invented at CERN; Google is just an integrator of European tech. Or Apple Silicon is British. ARM was designed in Cambridge; Apple just licenses it.