Mold is a plant, and it needs to be treated like an unwanted weed. Replacing the ceiling will get rid of what is growing there now, but it will just come back if you still have the same conditions.
To kill mold, I like to use Swimming Pool Liquid Shock. A gallon of it at Home Depot is just a few bucks, and it's in the outdoor garden area with pond and pool supplies. Put it in a spray gun and soak the area. This will kill the mold down to it's roots.
I'm also a big fan of Zinsser primer. In my experience, it's the only one that actually works.
Then I use exterior paint in bathrooms. Exterior paint is harder and seals better then interior paint. I don't think any of the mold blocking stuff does anything for interior paint. I buy paint from Sherwin Williams for most of my jobs, and usually get the more expensive stuff if my clients are willing to pay for it. Emerald Paint is probably the very best paint that you can buy.
To help prevent mold, the surface needs to be dry. Air flow is the very best way to keep everything dry. When I do a remodel for a client, I like to put the ceiling vent fan right at the entrance to the shower. Usually, it's close to the toilet too, but my main goal is to clear out the steam from the shower, so the ceiling remains dry.
To answer your question about what material is best to use for a ceiling in a bathroom, that would be the green Mold Guard sheetrock designed for moisture. But I would not replace the existing sheetrock unless it was damaged and needed to be replaced.
Most ceilings are 5/8's inches thick because the ceiling joists are 24 inches apart. It's not always done that way, and that's another issue, but if you have 5/8's sheetrock up there already, nothing else is going to have that thickness. If your ceiling joists are 16 inches apart, or less, half inch sheetrock can be used, or you can use something else.
The only reason to use something like GoBoard or Hardie on a ceiling is if you are going to tile the ceiling. And then you need to add lumber to the joists to get your span to 16 inches or less. I do mine at 12 inches when I tile a ceiling in a shower.
