I have fed the grass. Always pick up sticks off the yard and save them up for late fall/winter when we enjoy using the fireplace a few times a week. I used to stir the ashes into the compost pile, but then learned that it can alter the pH in a way that slows down decomposition. However, that same shift in pH is good for grass! it's the lime content. Last fall when it was time to clean out the fireplace I sprinkled a bucketfull of ashes on the back yard- to see what resulted. I swear this spring the moss is dying out and grass looks a bit better.
Just a few days ago did the same to the front lawn. My fireplace was full of ashes, I hadn't shoveled it out since the last use this past winter... It is not so easy to do, spreading the stuff on your lawn. Have to wear gloves- avoid skin contact. There must be no wind, not even a slight breeze- don't want to breathe it in, or have it blown away. Must rain (or get watered in well with the hose) to wash the ash off the grass blades and into the soil. And/or rake it in.
So I scattered ashes in the rain. I had my nearly-full five gallon bucket on the sheltered porch, an umbrella and a plastic scoop. Each scoop got scattered as evenly as I could over the grass, under the umbrella so my scoop wouldn't get wet and clump it up. Then I ditched the umbrella and raked it all in. Tedious, and I'm sure the neighbors wondered what the heck I was doing. I've seen a few people around here push a spreader, most folks seem to have a lawn service, I don't know anyone else who scatters their fireplace contents. (Which, by the way, can only have wood burned in it now. No cardboard, and very little paper- because only the wood ash is good for the grass). I think of all those scenes in nature programs that show the sweep of a wildfire, the blackened devastation, and then the flush of new grass, fertilized by the passage of flames...
My other change in methods is to mow at 4", every other week when growth is fast. The slightly higher grass is supposed to shade out crabgrass seedlings. I have always dug up crabgrass clumps when I could, and cut all the seedstalks before they scatter open. There was still always lots of crabgrass. But now almost a year after changing the mowing height there is far less crabgrass. I'm still digging up dandelions and other weeds by hand (although I leave the violets, plantain and clover patches). Also leaving the cut grass to fall on the lawn and feed it naturally- that's why I noticed my compost pile had way fewer greens- I'd been leaving them on the lawn.
I have a control, too. I'm not touching the hellstrip. It gets the altered mowing height, but no feedings of ashes or compost. I dig fewer weeds there, too. Place where I read about these methods said changing the frequency and height of mowing alone could solve 90% of your weed problems. Well, let's see. If the hellstrip improves too, that's about right.