Moss is there for a culmination of things...
Shade usually gets the blame, but it will also be lack of nutrition (we remove food all the time when we mow)
Low, irregular mowing.........Saturated soils........ lack of moss control.
Scarifying removes the plant but not the spores....
When using domestic products, the moss killer part cannot hit all plants, let alone spores, so try to do it AFTER you have scarified...
This way, you open up the sward so that the moss killer can get into the bottom of the sward.
Next time you look, when applying first is that you will only dis-colour the top part of the plant...and there will be many healthy plants ready to carry on!
Hollow tine aerate the soils as much as poss.....dont dress sand unless you can gurantee that the sand particles are the correct size and shape.
Leave holes open for roots to get in and develop.
Scarifying will also allow certain species of grass to thicken back up.
Best way to scarify....start as high as possible...and slowly move it down a tad....
Leave some grass to fill in.....after a week or so, apply a spring feed.....
Best to leave weed control as many of the new young recovery shoots may be affected..
Good luck