Just to emphasise, the making of firework powder is comparatively easy. Good firearms powder is another matter.
Ordinary charcoal is hardwood charred in a covered pit or chamber at a high temperature. Good firearms powder requires lightweight woods, charred in a temperature controlled sealed burn at a low temperature with the other products of combustion removed during the process but retaining the fouling softening creosotes. Then ball milled to an impalpable dust and it can then be used. Termed a meal powder.
Delivered sulphur contains assorted contaminants and good performance requires it to be melted and the light contaminants skimmed off or left and the heavy ones left behind as the middle liquid is drawn off, cooled, broken and ball milled as above.
The saltpetre is even more contaminated and needs to be put into solution and crystallised. Perhaps several times to get a near pure result. Then again ball milled etc.
After the same mixing as firework powder it is put into expensive powered rolling mills (weighing tons) under careful controls and mechanically incorporated under these many tons of pressure for several hours. The longer the better. Then it is dampened and pressed at high pressure until it forms a hard cake. Once dried out the cake is broken up and then milled into chosen size grains. Said grains aresubject to being rolled loose to glaze the grains. All the the dust is removed and returned to be used in the next batch. Only them is it fit to be used as dense smooth grains. There are variations in wood, temperature of burn, length of milling time, density of the cake pressing, the size of the grains and general variations in the quality of the ingredients including the water used to dampen the powder for milling and for caking. Today one can buy firearms powder of a period 'musket' standard, 'rifle' powder and rarely a true 'sporting' powder to match rifle standards as in the late 19th century. Even in those days they could be tailored to give burning at chosen temperature. In one case a proportion of the fine charcoal was mixed with 30% charred peas so as to reduce the temperature of the burn to prevent the fouling being too hard in certain cases. The modern Aubonne works uses more saltpetre in their powder than normal which reduces the gas production slightly but increases the adiabatic expansion of those hot gases to give a better and more constant burn.
I mention all of this to show how hard it is to make a good firearm powder of even 'musket' powder standard and needs a customer willing to pay a far higher price than the minimum and an industry willing and able to invest in the capital and skilled staff to achieve it. The British experience of their suppliers in the ARW led them to demand higher standards and to invest heavily in the means to make it thus. French powder, at that time, was far better and made in government mills built to make good powder. If it took Britain until the Napoleonic wars to make the change and I doubt if Mexico had either the means, the will or the government culture to do the same only 20 yers later.
I will mention that meal powder can be corned, badly, with lighter pressure and little milling, even in a stamp mill by simply grating of the lightly pressed, or just dried, cake and passing the gratings through chosen sieves but the result is soft and prone to crumble in storage and transport. In which case the powder will need to be remixed or good grains separated from the crumbled dust. Gun powder is a very complex substance from which to get the best results. By the end of the use of gun powder in small arms it was moving into solid rods with assorted shaped holes to adjust the rate of burn so there was a collective sigh of relief when smokeless nitro powders were introduced.
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