Suppose people are labeled with positive integers, and rooms are labeled with positive integers.

Suppose first the people labeled with even integers show up, and they each go in the room with half their number. Then each room will have a person.

Then the person with number one shows up, and none of the rooms are empty.

If there are א people that exist, not all collections of א people have all people.

I think I don't have an intuitive understanding of how your intuition works.