I don't recall if it is just by handshake agreement or in an actual treaty but the country where the manufacturer of a plane operates is the one designated authority over that manufacturer.

FAA issues Airworthiness directives and sets rules Boeing has to follow. EASA largely defers to the FAA (not always but in almost everything).

Similarly the FAA follows EASA when it comes to issues with Airbus.

Both FAA and EASA requirements are roughly alined anyway so it doesn't make that much difference.

Boeing has been coasting on their engineer-led culture from the past. They're only allowed to self-certify on things that are not "safety critical" so having the FAA review that designation seems more useful.

FWIW when Boeing merged with McDonnel-Douglas the GE protégé crew took over. The decision to move headquarters was done deliberately so the machinists and engineers couldn't go complain to the bean counters in person.

Also think about Boeing setting up a factor in South Carolina and fighting so hard to prevent that factory from unionizing. They're focused on giving labor a smaller share of the pie... on some of the most expensive (and formerly profitable) things humans build, with a decade-long order book where every single unit they make is pre-sold years in advance. (Recent screwups have hurt them badly though).

Do you want an airplane built by an accountant-run company that cares about finding the cheapest most disposable workers to build the airplane? Whether it matters or whether the union sucks is irrelevant. The fact that Boeing management was so determined to screw the workers who assemble these planes says a lot and none of it is good.