Fire is much more complex than that. Fires appearance comes from:

• blackbody radiation, as you point out from smoke particles but also from gaseous components
• chemiluminescence, the chemical reaction itself emits light, and this is why fires can burn in different colors. In fact you can buy additives which are generally metals which make fires burn blue or green or red, etc.

Fire is an active chemical reaction. It's a transition between often solid or liquid, sometimes gaseous, fuels, into gaseous products, all while undergoing a chemical reaction. It's not a state of matter, states of matter concern the phase of equilibrium conditions, and fire is decidedly not in equilibrium.

"With this in mind, it should be clear that a candle flame gives off light even though it is not a plasma. In contrast to candle flames, certain burning mixtures of acetylene can reach 3,100 degrees Celsius, with an associated Debye length of 0.01 millimeters, according to the Coalition for Plasma Science. Such flames are therefore plasmas (as long as the flame is much larger than 0.01 millimeters, which is usually the case). Other flames, including flames from campfires, propane stoves, and cigarette lighters, have temperatures that lie somewhere between these two extremes, and therefore may or may not be plasma."

https://www.wtamu.edu/~cbaird/sq/mobile/2014/05/28/do-flames-contain-plasma/

I somehow never knew what "fire" was until reading this comment. Huh.