BUENOS AIRES—In the aftermath of a deadly terror attack that stirred Americans’ concerns about the potential for threats to the U.S., President Barack Obama pressed ahead with his tour of Latin America, including a planned family excursion in Patagonia.

Mr. Obama’s public appearance of nonchalance has drawn criticism from Republicans that he is detached from Americans’ fears and isn’t sufficiently countering violent extremism. But his approach partly reflects his belief that overreacting to a terrorist attack—however horrific—elevates...