The CBEF test is worth 250 out of the total 1400 points. To be honest I don't think there are a lot of applicants that fail or score low enought on this test to dramatically change their score.
The reason your son was advised to better his PFT score before completing his application is because once an applicant is Boarded, the applicant is not allowed to update the PFT.
The CBEF was most likely not the reason your son did not receive a scholarship. The Whole Person Score contains three categories. Academics, Leadership, and Athletics, think of it as a three legged stool, if an applicant does not have one of those legs then the stool falls over. Your son had great Academics, the first leg. He had leadership in JROTC, not sure what other leadership he had, JROTC in itself would not be enough. Your son devoted his efforts to JROTC and did not have the school Athletics, that is most likely the leg of the stool that was missing. While his GPA and SAT were terrific, they were only one leg of the stool.
Every year there are applicants that receive a scholarship with a GPA of 3.4 and a SAT of 1100. These applicants were very heavy in Leadership and Athletics, many were Captains of their teams and had multiple Varsity Letters, they are in student government and several leadership positions. The minimums for Academics to achieve that third leg are not that high when an applicant has two other solid legs to back it up.
Not having Athletics also prevents an applicant from getting 100% on the interview scoring which can effect their chances of getting a scholarship as well.
As someone posted earlier, the majority of cadets in ROTC are not on scholarship when the join ROTC. Some will get campus scholarships along the way, some will join the Guard and do SMP, and some will complete ROTC, graduate, and commission into Active Duty having never had a scholarship. These cadets find a way to make it work if their goal is to become an officer in the Army.