It doesn't make sense to require psych testing for every admission. Many patients have already had testing prior, so testing again is redundant. Others may only need a few days of emergent care while getting services put in place, and then their own therapists or psychiatrists can arrange for testing later when they are not in an emergency situation. Testing requires hours of evaluation that is not really feasible in a hospital situation where there may be dozens of patients, many of whom won't be there more than a few days.
IF a patient is put into long-term hospitalization, then of course, testing should be done. Most inpatient psychiatric care in the U.S. is short-term crisis management, however.