An oxy-acetylene flame can reach 3500°, which can easily heat steel past its kindling point of about 870°. Other oxy-fuel flames, like propane at 2800°, can also flame-cut steel. But single-tank torches that mix their fuel with air generally cannot, because they have to heat up a lot of inert nitrogen along with the active components, and even though the resulting temperature is higher than steel’s kindling point, it’s not hot enough to heat the steel fast enough; air-acetylene tops out at 2500°, propane at 1980°, gasoline at 2140°, hydrogen at 2250°, wood at 1980°.
You might be tempted to heat the air by running it through two flames, but that doesn’t work, because the oxygen in the air has been used up; you’d have to mix oxygen back in, which lowers the temperature. What you need is a way to separate the heat from the gas and transfer it to fresh air, which you then use to burn a second flame.
You can do this with a regenerator. Taking propane as an example, you can heat the regenerator to 1980° with a first propane flame, and then heat fresh air to 1980° with the regenerator before using it to fuel a second propane flame. Propane has a specific heat capacity of 73.6 J/K/mol (and a molar mass of 44.1 g/mol), while air’s is about 29.2 J/K/mol (and a molar mass of basically 30 g/mol), the stoichiometric mixture is C₃H₈ + 10O₂ → 3CO₂ + 4H₂O, and air is only about 21% oxygen, so each mole of propane needs 10 moles of oxygen and 47.6 moles of air to burn. So the stoichiometric mixture is 97.9% air by volume, 97.0% air by mass, and 95.0% air by thermal mass, so if the air is at 1980° and the propane is at 20°, then the mixture would be at 1882° if it didn’t immediately burn.
But it does immediately burn, which should raise its temperature by another 1960 K to about 3840°, quite a bit hotter than the oxy-acetylene flame, and plenty hot enough to cut steel.
For such a high temperature, both the regenerator and the combustion chamber would need to be of a material that can withstand the full 1980° produced by the propane-air flame. This is fairly demanding, and only a relatively small number of such refractory materials are available, including zirconia (melts at 2715°), urania (melts at 2865°), thoria (melts at 3300°), graphite or carbon (subliming at 6000°, but will burn at 700° if oxygen is available), and lime (melts at 2615°.) But if we only want to achieve the 2800° achieved by an oxy-propane flame, the regenerator only needs to withstand 840°, which is achievable by any number of everyday materials, including iron, steel, copper, and most kinds of dirt and rock.
It isn’t necessary to use modern fuels like propane; syngas from a wood gasifier would work just as well.
(Other low-tech ways of flame-cutting steel are possible; Theodore Gray has demonstrated flame-cutting steel using thermic lances made of prosciutto and breadsticks and a cucumber, for example.)
There are several ways you could arrange the regenerator cycles; for example, you could have two or three regenerators and alternate between them using valves, or you could mount the regenerators in a wheel like the desiccant wheel of a desiccant-type dehumidifier.
This approach should make it possible to flame-cut steel without using any exotic materials.
If we use the regenerator technique to optimize for high temperature rather than for easily-available materials, using a thoria regenerator to preheat oxygen for an oxy-acetylene flame should enable much higher temperatures, though not the 6800° you’d naïvely think, because the acetylene is a much larger fraction of the thermal mass entering such a flame, and the acetylene is by necessity not preheated.
This approach should make it possible to flame-cut stainless steels and aluminum, maybe glass and stone and other materials that cannot normally be flame-cut except with a plasma torch. (Plasma torches can reach over 20,000°.)