When repairing water damage to our ceiling from rain blowing through our louvered gable, does this need to be cut out and replaced?

Stained drywall
Stained drywall may not need to be replaced. If it is only stained from a one-time event, it is still structurally sound. Only if it is sagging down, which can happen if it were super saturated, would I do this.

Otherwise, scrape any loose texture or paint, fix any bubbling tape seams, and paint the stained area with a stain-blocking primer. Kilz and Zinsser both make products for this purpose. If it is a bad stain, two coats are a good idea, but be sure the first coat is dry before applying the second. At that point, you can retexture, or repaint the area to match. Painting the entire ceiling may be required to tie it all together, so it does not look like a repair.