I don't understand how someone could take her son's pride in his service away from him, so I certainly don't see how someone could take her pride in his service away from her.

But where are you getting "pride" from? I haven't heard the speech, so I'm just going by your quote above. And according to your quote, she said "you can't take that away from me" (emphasis added). The most natural antecedent for "that" is "having raised a combat vet." To me, that implies the whole experience, not just her feeling of pride in her son's choice of career path. I think you're being too narrow in your verbal analysis.