Bob, the first successful commercial DA/SA was the Walther PP, Polizeipistole or Police Pistol which was introduced in 1929, nine years before the P38. The PPK, Polizeipistole kriminal or Police Pistol Criminal for detectives, was introduced in 1932. So only three years after Browning passed for the PP to be introduced.
It's just that it seems to me, if he'd have put his mind to it, he could have done it. Part of the reason he pursued the HI Power, was because he wasn't completely satisfied with the 1911, even though it was a great success as a handgun and continues to be so. I've looked at and studied his patents, and as innovative as he was, the lack of a DA/SA design has me wondering why. The extra work in both the 1911 and the HI Power to make them capable of carry in condition one or cocked and locked, seems to be where he ended his pursuit design wise.