All anesthesia providers are supposed to do a pre-op H&P that is recorded on the anesthesia record. That H&P is what's used to determine the Physical Status score (P1-P6). So, yes, the anesthesia record is used to determine the ASA score.
Often times, the surgeon will not document co-morbidity diagnoses that appear to be irrelevant to the case, however those diagnoses are important to the anesthesia service. The surgeon's documentation demonstrates why the anesthesia service was necessary, and the anesthesia record should contain documentation proving the medical necessity for the level of anesthesia service provided.