Because reasonable APIs have input validation provided out of the box, so that not everybody has to reinvent the validation wheel (inevitably incorrectly).

Parsing user input? Nonsense data coming from an API?

It's mainly horrid, because it means you have to code extremely defensively (or I guess, use a different API).
You can't rely on new Date("not a date") aborting execution of your function by throwing an error. Instead, you have to know that it can produce an Invalid Date object and check for that. Otherwise a random NaN shows up during execution, which is gonna be extremely fun to try to find the source of.

I understand that it's implemented like that partially for historical reasons, partially because it's often better to display "NaN" rather than nothing, but it's still the sort of behavior that puts me in a cold sweat, because I should be memorizing all kinds of Best Practices™ before trying to code JavaScript.

You think you'd get an error from the constructor.

Because this is a simplified example? Maybe you create the object in one place (saying something more realistic like "2015" or whatever your inexperience or AI told you to) and use getTime() at a later place where you thought you created it in a correct way.