The vast majority of sales are made to US based firms so they likely have a lot of sway.

The sway is TSMC uses ASML EUV lithography machines and the US holds patents on those because they did foundational research regarding EUV lithography. Also, the EU hasn't put China on the "it is illegal for EU companies to kowtow to US sanctions" list. Ironically ASML could sell to Cuba and Iran. If the EU were to tell ASML to sell to China the US would be free to not buy ASML machines any more and, doing that, kill off Intel's fabs.

None of this stuff has military relevance, you don't need or even want to use small nodes (which require EUV) in military applications you want hardened chips instead. Run off the mill consumer chips go all frizzy if an EMP looks at them sideways. This is about the US protecting US fabs, foremost Intel. Not the chip design part but the manufacturing one.

Europe hasn't played the high-end end-consumer chip market for ages and I doubt we'll do it any time soon. Having ASML, Zeiss etc. means that whoever actually produces that stuff wants to be friendly with us and strategically, both military and economy, our own production facilities are perfectly sufficient. Hence also why ESMC will only go as small as 12nm, it's the most cost-effective node size and performance is perfectly adequate for a missile, a CNC mill, or a car infotainment system. Or the gyroscope chip in your phone (it's almost certainly a Bosch), EUV doesn't make a lick of sense when you're doing MEMS. Where we have to catch up is chip design lets see how that RISC-V supercomputer chip turns out.