How do we fix this ugly chimney?

Looks like it needs professional sand blasting to me.

1. Mix one cup of bleach into a gallon of water.
2. Apply the solution on a small sample area first, then wait a week to see if there are any unwanted color changes in the brick.
3. If there are no unwanted color changes, thoroughly soak the bricks with water.
4. Using a spray bottle or hose spray attachment, spray the bleach solution on the bricks while they are wet.
5. Let it sit for a week, then thoroughly soak the bricks again with water.
6. Remove the moss with the scrub brush.
7. Rinse.
8. If this does not get all the moss, use a power washer to blast the rest off.
9. Better results are obtained when the moss is actively growing, but you should be able to remove the stuff even in cold weather. Avoid tightly focusing the spray or using so high a pressure that you damage the brick; start low and work your way up, keeping the pressure well below 3,000 psi.
10. Always rinse thoroughly with clean water.

Be sure to wear rubber gloves as you scrub, and don’t forget eye protection—scrubbing and spraying can splatter the bleach mixture toward your face.

Why not cover it with a stacked stone or something else of your choice? When done right, people will think it's been there all of the time.

I agree that sandblasting is the best way to clean it up.

How did that even pass code? Sandblast it or even painting will make it look better.

Sandblasting is a process. Different products are used in sandblasting based on what the item is being blasted. It's not just "sand". A lot of auto shops sandblast paint off cars using baking soda, ground up pecan shells, etc. Graffiti on brick and concrete is sandblasted off rather than painting like they did years ago.