They are complimentary. Shutters reduce the amount of heat coming in. AC gets whatever heat managed to go in out.

One thing I'll never understand is not having shutters, only curtains. I live in France, almost everyone has shutters. But in other countries it's often different. I lived in Thailand for a couple of years and had only curtains, so the sun would still get past the window pane, heat the curtain, and the AC would then have take this heat outside again. This seemed massively inefficient.

I thing the best compromise is using whatever traditional, passive means of cooling we can, and then AC as needed.