I'm sure the actual legal mechanisms are complicated, but I don't see any justification for the US imposing export controls on technology developed in-house by a European company.
It's possible the scenario is something like this: American company XYZ has key patent on feature ABC that is required to build this type of hardware. ASML licenses ABC patent from XYZ. If they sell the resulting product to the Chinese, then the US comes after XYZ for violating some sort of export law that prohibits granting IP licenses to people who sell to the Bad Guys. So XYZ is more or less forced to include export-control language in their patent license agreement.
There are probably several such guns aimed at ASML's head. The US is really, really good at coming up with things like that.
You mention there probably several such guns aimed at ASML's head and I have no idea what set of carrot+sticks are involved but want to point out it might not just be sticks/guns.
It might additionally be an appeal to ASML or Dutch self-interest. China does not have a good reputation for respecting IP (and I do not think this is just propaganda.) ASML is not somehow exempt to that medium/long-term in one way or the other and has already had issues with that vis-a-vis China (https://www.cnbc.com/2023/02/15/critical-chip-firm-asml-says...)
Sometimes a corporate entity like ASML needs government(s) to take a stand so that there is a more level playing field so they don't lose out to local/smaller competitors who take a different approach.
ASML is capable of deciding for itself whether it's risky from an IP perspective to sell to company X or Y. It's not ASML that is asking for these export controls. In fact, ASML executives have been warning that these export controls will spur Chinese investment in indigenous chip-making equipment providers, threatening ASML's business over the long term.
I completely agree that this is a significant risk of the export controls.
But China was already expected to climb their way into semiconductor manufacturing with significant investment. What, were they not going to do that? ASML was threatened long-term either way.
But I agree it will definitely focus minds in China on the problem and lead to some acceleration of their progress.
It's possible the scenario is something like this: American company XYZ has key patent on feature ABC that is required to build this type of hardware. ASML licenses ABC patent from XYZ. If they sell the resulting product to the Chinese, then the US comes after XYZ for violating some sort of export law that prohibits granting IP licenses to people who sell to the Bad Guys. So XYZ is more or less forced to include export-control language in their patent license agreement.
There are probably several such guns aimed at ASML's head. The US is really, really good at coming up with things like that.