Jesus used unleaved wheat bread. He didn't use cake. He didn't use barley. He didn't use peanut butter. He didn't use fried chicken. Your quarrel is with Jesus, not the Church.
Jesus did not say our unleavened bread had to be made from wheat or it would not be accepted, if it had been an order he would have clarified it....he did not. So therefore 'unleavened is key' not the flour used.
This sort of stuff is part of the reason I realized Catholicism was not for me, and converted to my fathers religion of Judaism.