In the US, you register with the party no? So the party you register with your information, and they can do what you said
> go to their houses, send them things, and know for sure they're only hitting up people in their party, so as not to mobilize the other side.
You can register to vote and self-identify as 'Dem' 'Republican' etc while getting your driver's license.
Aside from the public voter file update, the Democratic Party doesn't get any special notification if you pick them, and you don't need to apply or get accepted in any way.
My guess is that it's public to avoid unfairly preferencing the major parties, so that they give at least lip service to the idea that "anyone can run for office". The U.S. is pretty sensitive about seeming like it has a fair and open political system, even if other aspects of the system mean that in practice a third party doesn't have a snowball's chance in hell.