ASML is European and is arguably the most strategically important company in the entire semiconductor supply chain.

ASML Holding is dominating the chip technology with their machines. It's not the lack of invention or intellectual capability that holds Europe back from the digital industry, it's the lack of willing long term European invenstors. If you want to scale your digital tech startup in Europe the most viable way is to look to the US for investors.

ASML is not a chip maker, it is a chip-maker maker. Still important though.

Europe should request a discount for ASML machines in a EU factory.

The hardware is in europe. The IP and control is in USA

I think that it would require there to be a European chip demand. Today that demand is almost entirely for cars, so we only get mediocre car infotainment chips (+ a few other similar niches). There was more hope 20 years ago, when there were widely successful European mobile phone makers.

To be fair, there is ESMC now which is being built out in collaboration (read major involvment) of TSMC.

https://www.esmc.eu/en/index.html