The early patents show that almost all the pistol designers tried using standard revolver cartridges, including Browning. The problem was that they needed the rim for case support, yet the rim made feeding from a magazine unreliable. Browning first took the approach of trying to make the rim as small as possible, giving us the "semi-rimmed" .32 ACP, 9mm Brownng Long, .25 ACP, and .38 ACP. Then in 1904, he found out about the 9mm Luger, a case supported on its mouth, and the light went on. His next two cartridges were the rimless .380 ACP and .45 ACP.
The 9mm Luger, in fact, came about by accident. Its origin was the 7.65 Borchardt, shortened to function in the Luger. It was supported on the case shoulder. But the German army wanted a 9mm, which meant trying to expand the case until there was almost no shoulder, not enough for support. So Luger hit on using the case mouth and that was the answer.
Jim