Administrators and housing officials would have no idea who is on and who is off campus at most schools.
There are many fine schools ranging in selectivity from UTAustin and the University of Michigan to Youngstown State University where you could be in “off campus,” non-university housing but situated just as conveniently (and in some cases more so) as the school’s dormitories. Then there are other schools like the University of Kentucky where first year students have easy access to apartment-style even living in school housing.
My kids both attend schools where it seems to be pretty common for some international students to choose apartment-style living immediately, and there is a myriad of personal, religious, and cultural reasons why they make this choice. These students become active, integrated members of the campus community, so if that is the kind of housing you want I would do the additional research to fine those kids or schools. My guess is that, in general, you should look for more urban and larger universities vs small, isolated LACs.