A BDA period IPSC shooter developed a reload that was not a whole lot slower than a Luger button*.
He hooked the exposed front of the magazine floorplate with his left forefinger, thumbed the latch, and PULLED. Then inserted a fresh magazine. He said he did have to walk a ways to get his mag back, the vigorous pull tended to fling it.
*If you look at the old guns, you will see that the first side button magazine catch on a mass production pistol was on the Luger, ca 1900. Early Colt/Browning designs had heel catches.
I have never seen it documented but would not doubt but what the Army liked the Luger arrangement and passed the word on to Colt between 1907 and 1909. Those being the prototypes that bracketed the change.
So the "American" magazine catch originated in Germany, and the "European" magazine catch was in common usage in the USA. Kind of like Boxer and Berdan primers, a prophet is not without honor save in his own country.