Your online research is correct. Napalm relies on oxygen in the atmosphere in order to burn, so it produces just an ordinary flame.
Ordinary flames reach equilibrium at less than few thousand degrees as radiation and convection carry away the limited energy output available from oxygen travelling towards the fuel.
A traditional flame is an "oxidation" process which releases energy by replacing oxygen double bonds with lower energy bonds. When the fuel is a hydrocarbon, as it is with napalm, oxygen is available only from the atmosphere. While the fuel is concentrated in a solid/liquid state, the gaseous oxygen in the atmosphere takes it time to get to the napalm. This is a major factor limiting the temperature of flames.