So matchlocks and flintlocks were used at the same time. The flintlock has a lot of benefits over the matchlock. It also doesn't seem to be that much of a technological advancement or challenge to make since lots of people had flint and steel so why did everyone just not immediately switch out matchlock guns for flintlock ones?
To be precise, there were a few ignition systems . There was first the matchlock, then the wheel lock appeared circa 1500, then soon after that the snaphaunce, a kind of flintlock. The wheel lock used a piece of iron pyrite held against a serrated wheel to make sparks ( those sparks were from the pyrite, not the wheel). The snaphaunce worked somewhat like the later flintlock- a hammer or cock struck against the steel or frizzen, making sparks. It was a little less surefire than the wheellock, and it had a pancover that had to be flipped aside, either ( first) manually or (later ) automatically.
The flintlock as we know it is largely the invention of one man. Louis XIII was quite a gun collector, and he installed a gunsmith/metalworker/inventor named Marin le Bourgeoys at the Louvre to work for him. Marin le Bourgeoys made some key improvements to the snaphaunce. Around 1610 he combined the frizzen with the pancover, so that when the flint hit the steel, it flipped open the pan. He added a half-cock notch to the tumbler, so that the gun could be safely carried loaded ( unless, yes, it went off half-cocked) and - a little arcane, but important- he changed the sear to disengaging vertically from the tumbler, instead of horizontally. That made it possible to have a shorter, lighter trigger pull ( as well as the half-cock).
The geometry for all this ( the length of the hammer, pivoting length and width of the frizzen, curve of the frizzen face, strength of the frizzen spring, strength of the main spring, etc) - was somewhat finicky, and had to be worked out. It took a bit of tinkering , after 1610, to get all that just right. And it is still somewhat finicky. If you talk to a hunter who uses muzzleloaders, both percussion and flintlock, they'll tell you that even a cheap percussion lock will work fine, but a flint lock has to be well made, and so will be more expensive if it's to be reliable.
The usual driver for innovation in weapons is warfare, and it was the Thirty Years War that seems to have made the snaphaunce obsolete and the flintlock more common, and the flintlock would mostly displace the quite expensive wheel lock by around 1650. But the simple, relatively cheap matchlock musket would still continue to be used to around 1690.
Torsten Lenk: The Flintlock