First of all, when most people refer to the "Privacy Act," they mean HIPAA. HIPAA involves privacy of medical information. Thus, BGOs and others not directly involved with healthcare issues cannot discuss medical issues with you. This is why BGOs CANNOT help candidates with DODMERB problems.
Despite the fact that the Supreme Court has found a "right to privacy" in the Consitution, I'm not aware of any "Privacy Act." Certain states and the federal government may have individual laws intended to protect certain personal information such as medical data and SSNs, but this I'm not aware of any prohibition on USNA releasing SAT scores to individuals who are working on behalf of USNA.
Your BGO does NOT have access to your application. In fact, quite the opposite. We get a limited amount of information, which includes your SAT/ACT scores, which items of your packet have been received by USNA and when they were received, your medical exam status (incomplete, qualified, rejected), where you go to school, whether you attended NASS, etc.
The reason we get your SATs is to help us counsel you. For example, if you have 600/620, I'm probably going to suggest you retake them. Your score does not affect my evaluation of your overall candidacy. We're not here to evaluate your grades or SATs -- USNA is perfectly capable of making its own assessment in that regard. We focus more on the intangibles -- motivation, desire, leadership ability, communications skills, etc. -- things USNA can't easily measure because they don't meet you other than through us.
Many of my candidates voluntarily bring their essays and/or resumes. Personally, I find it very helpful but I do not suggest or require they do so.
BGOs exist primarily to help students -- well qualified and not -- on their admissions journey. USNA actually provides us almost no information about the candidates (for example, we don't know how well you did on your CFA or even if you passed), which actually makes our lives more difficult at times. They do this out of respect for your privacy.
So, to summarize -- we do NOT have access to your on-line application. We receive a very limited amount of information about you that is intended to help us help you with your application. If you choose to provide nothing more, that's your prerogative, but in the end, you're the one who is likely to lose out.