Not sure what the issue is?

Canada is the same thing with their ETA, where Canadian citizens can not get an ETA (though to be fair, US citizens don’t need one either). Im a dual US and Canadian citizen - when traveling north to Canada I check in and use my Canadian document, when I come back south to US I use my US documents. Been doing this my entire life, and never an issue. So I fail to see why you couldn’t do the same thing for a dual US-EU citizen.