First, I am not Catholic but Jewish. First Things is not Catholic (although its founder was Catholic) but ecumenical.
Second, I support Just War doctrine. I supported the invasion of Iraq but thought it stupid to take responsibility for nation-building in Iraq. We should have put in our own strongman and withdrawn most of our forces.
Third, Petraeus solved Bush’s problem in Iraq brilliantly, but at the cost of creating an effective Sunni armed force, which will create trouble in the future.
The whole nation-building thing was a very risky process, but if you do pull it off the dividends can be very long-lasting. Kenya and India come to mind. Whether Iraq will turn out that way we cannot know yet. But I'd submit the historical record suggests that if you do pull it off, it's longer-lasting than any strongman and quite a bit better for the population of the country it's done with. Personally I admire Bush for sticking to that after the staff who came up with it started to peel off. YMMV.
No doubt he received the AEI invite prior to his statement to the committee.