If your charcoal is actually pure wood charcoal then it seems that there is no debate - that is wood ash and adds potassium - think of how a forest does well after a fire from the charcoal added, and do a search on biochar. Gardening buffs seem however to have a problem with many of the petrochemicals added to most commercial charcoals that make for easy lighting etc. as well as the sulfer content alluded to above.
That said I dug my left-overs from the regular stuff in my garden years ago and it caused no problems. Don’t know if it help much though. I would dig it in. Without that it tended to flow off in the rain onto the walkway.
composting is letting organic matter partially decay. it is good to do with grass clippings and leaves. kitchen waste that is vegetable only (no meat, dairy, eggs, oil, fat) is not stinky or attracts vermin.
wood ashes or wood charcoal ashes are alkaline. many plants like soil to be slightly alkaline. pines make soil acidic (leaving the needles on the ground does). charcoal briquettes may have clay and other materials in them.
adding compost, sand, black dirt or ashes will loosen clay. you need to rake or till into the clay. you can place these on top of soil and rake in with a garden rake, even small amounts through existing grass. you can also make or put soil on top of clay though you need a few inches to do that.
We don’t compost briquette ash since it has added accelerants and fat drippings. We do use fireplace ash that’s all wood. Lilacs and some other plants like a more alkaline soil, as noted above, and you can change the color of hydrangea blossoms.
More than you asked for: We also live in a high-clay area. This is tedious, but I do a section of the yard every year:
Clear weeds.
Dig out the clay in a 2-foot wide trench to 1-2 feet deep (we do closer to 1 because we’re on a pretty steep slope).
In a deep wheelbarrow, mix chopped-up clay, bark mulch (shredded, not nuggets), compost or leaves/grass clippings, screened topsoil, a few cups of dolomite.
Fill the trench half-way and add organic fertilizer.
Finish filling trench.
Dig next trench.
Top-dress with ash where desired.
Top-dress with bark mulch.
Plant and water.
Another strategy I’ve used successfully is to turf off the grass, chop up the clay a little and sprinkle with dolomite, then cover with compost and bark mulch. Leave it alone for a year, after which it’s easier to dig or may be amended enough to plant as is. On a hill, adequate drainage isn’t a big issue, though in flatter areas, make sure you’ve chopped any planting holes enough that water won’t just stand in the hole. If it does, you can try running a French drain or adding a layer of gravel to the bottom of the trench.
ETA: Our yard has such high clay content and is steep enough that tillers don’t work.
In small quantities, charcoal ash would be OK as a soil amendment, especially if your soil needs to be of a higher pH (such ash is alkaline and raises soil pH). It would probably not be a good idea around plants that like an acidic soil (i.e. rhododendrons).
The best amendment for a clayey soil is organic matter, including fine mulches that will decompose and gradually improve the soil. As to what a previous poster said about pine trees, I don’t think they generally tend to make soils around them more acidic (pine needles are said to minimally affect pH when worked into the ground).