It's not a flawed premise! It's bad wording. The premise is, given an inelastic supply of housing, more immigrants means more of the supply occupied by them,

It's not a flawed premise! It's bad wording. The premise is, given an inelastic supply of housing, more immigrants means more of the supply occupied by them, leaving less of the supply available for non-immigrants. Point out a flaw in that, if you see one.