Soot is amorphous carbon; that will absolutely be burned at higher temperatures (given sufficient oxygen). You might be thinking of ash, which consists mostly

Soot is amorphous carbon; that will absolutely be burned at higher temperatures (given sufficient oxygen). You might be thinking of ash, which consists mostly of inorganic residue (although even there you'll see carbonates at low temperatures and metal oxides at higher temperatures). But when we talk about burning wood cleanly we're really talking about smoke -- and that's almost entirely organics (including large amounts of carbon monoxide, methane, and VOCs) which will combust given sufficient heat and oxygen.