True, the counselors at the private schools aren’t able to overturn an admissions decision. But they do have a lot of influence. They have unofficial teleconferences with the admissions offices to review the slate of candidates. During these calls, any small comments could tip a candidate over the edge.
My son attended a boarding school whose entire staff of college counselors were former directors at elite/Ivy level colleges. Listen carefully to what they say. The messages are always indirect.
Reading through the tea leaves, it appears your son will be facing competition from within his own class at these top colleges. So thats why they told him everything is a reach.