If you look at the firearms of the 19th century Qing army, you mainly see matchlocks without paper cartridge or socket bayonet. Basically the equivalent to the European early 17th century. Which is as it happens is the tail end of combat archers in Europe. The last battle with archers in Europe was iirc in the English civil war. Of course by that time archers were mainly used in place of light artillery, a field were early 19th century Qing China actually had some modern pieces, but underdeveloped doctrine and training. Even so the Qing were increasing the share of gunpowder weapons
That indicates to me that the 19th century Qing themselves were technologically right at the tipping point where the advantages of the firearms (ease of training, density of formations, ease of logistics, robustness, flexibility, sustained firepower) heavily start to outweigh the advantages of bows (indirect fire, somewhat reusable ammo, range and rof under good conditions). Of course European armies of the 19th century were well past that point