> imagine if in The Netherlands you can waltz in and say you are this person or that person

That's not at all how it works. Typically, you do need to present ID to register. You also need to verify your address on the day of voting. If someone does commit fraud, it's easy to detect: as soon as the real voter tries to vote, the system will detect a "double vote." The fact that this almost never happens suggests that there is not much fraud, if any. And the amount of fraud required to actually change an election result with this method would require a concerted effort of tens of thousands of people, all risking federal prosecution.

> that is literally the American system.

No. Please stop misrepresenting the facts.

> But why not just... require ID to vote?

There are lots of good reasons, which you are probably already aware of, including the fact that many people eligible to vote do not have the requisite ID and are unlikely to get it. My litmus test for supporting voter ID is that it is accompanied by (a) free, mandatory, and automatic national ID cards, and (b) national holiday on election days. Republicans won't support part A for their own ideological reasons, and moreover their purported effort to support voter integrity is not sincere: the goal is simply to prevent undesirable (i.e. non-Republican) voters from voting.