Is the idea of "boarding university" still a formal policy?
In the United States, it tends to depend strongly on the particular university and the population that it aims to serve.  In general, the more local and "non-traditional" that students are expected to be, the less that a university will feel obligated to assist in the arrangement of housing and board.

Thus, for example, a place like Williams College, where the students are mostly "traditional" undergraduates coming from far away to a small isolated town in the mountains, expects to house, board, and provide a full campus life for essentially all of its students.  

At the opposite end of the spectrum there are places like Suffolk University, which is in the middle of a big city (Boston) and historically focused strongly on professional postgraduate education and on students enhancing their education later in life.  Suffolk doesn't even have a campus per se---it just has about a dozen buildings scattered around the center of Boston, and until quite recently did not have any dormitories at all.

Across the whole spectrum, most administrations will consider it their responsibility to provide a rich intellectual environment for students both inside and outside of the classroom, e.g., by supporting student groups.  The degree to which this happens "on campus" and with direct university investment varies greatly depending on circumstance.  No university in America, however, will attempt to confine students to a campus---some merely recognize that geography makes it more difficult for their students to leave.