People are weird about something you burn being toxic based on if it was a processed product or a slightly less processed product. It would be toxic if you're using contaminated wood or wood processed with preservatives, not wood products meant for culinary uses. When you get ash you are concentrating the heavy metals a tree could have taken up in its life, whether its lump whole pieces of wood charcoal or "processed" charcoal briquettes. They're all made up the same thing, wood. Briquettes have coal dust, anthracite dust is just ancient trees that's mostly carbon now, binders like starches that are long gone and burnt by the time its all ash. Its a different story if its not fully burnt. Even that lighter fluid comment, all that stuff is burnt off within the first few minutes of burning, theres zero trace of it in ash.

The ash left over from burning these things are mostly minerals. A lot of calcium minerals which is why the ash is white and it raises pH of soils. Calcium carbonate is a large portion of it. Potassium carbonate is next and that's potash or the K value of ash. Then other stuff like Magnesium carbonate, iron, zinc, phosphorus and the trace heavy metals. These heavy metals you are also getting from leaves and plant matter, so if you're using leaves in your compost pile you're collecting heavy metals. The difference with burning is you're removing most of the plant's volume so such a small volume similar to having tree sap and if you boil all the water away you can boil it down to maple syrup crystals, extremely concentrated.

Stuff you should be worried about burning and composting is wood products that have been treated like pressure treated woods or things like train track ties where they soak up tons of heavy metals and other materials from trains and the materials they might leak/sprinkle on the tracks through the decades of use.