The example you mention is a tricky one, ASML is about as European as it is American. It is heavily subject to US export controls because its machines include US components, US software and US IP. They operate multiple R&D centers and factories in the US and employ a lot of US employees (~20%)

Not just ASML, when I working in EU at another major Dutch semiconductor company, the leadership and decision making was mostly US-based and centered around US policies, even if the HQ was still technically in the NL, but because PE and most other major investors were all US Bay Area.

For example my manager had to reject a candidate from Iran, simply due to US policies, even if that wasn't against the EU policies, and would actually have been illegal discrimination in the EU where the position was opened, but this policy came from the US and was applied in an unwritten fashion in the EU.

The influence of the US on finance and tech is massively understated. Plenty of international EU tech companies whose business model doesn't revolve around catering to the EU market and sovereignty, will tailor their products and operations to comply with US regulations since that's where a lot of their customer base and money is made. Just read the last post of the CEO of Bird, you can disagree with him that he's a scumbag and he's full of shit, but the truth is if you want your SaaS to make money, you gotta be in the US. That's why Mistral has a large presence in SV and London. Continental EU just sucks for acquiring capital.

Even at the no-name Austrian startup I used to work before, the biggest investor was a US PE, since no German or Austrian investor wanted to invest more than 5 figures in the company, which was not enough to cover the payroll of the ML/data scientists. As long as the EU investors refuse to invest in domestic companies and their future, EU companies will forever be tied suckling to the US teat if they wish to grow and survive.