abstracting this claim to

["my X"]:

rest of the speech

then we see this rhetoric style from the start of the republic and it continues. A very simple way of showing this is to just look at all the presidential inaugural addresses.

This literally dates back to Washington's (thus since there this was a president) with "my fellow citizens" or "Fellow-Citizens of the Senate and of the House of Representatives" (how his first inaugural starts).

Early American political rhetoric isn't my wheelhouse so i'm not going to do a deep anlysis of this specifically but it's good to note that the creation of the specific phrase you're asking about is separate from the rhetorical cadence and structure that the words fill in.

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the first president to start speeches with literally "my fellow americans" appears to be Calvin Cooledge in a 1927 address to a joint session of congress but the term becomes etched into the American consciousness because of FDR's use of it.

While his 1st innaugeral starts with the line

I am certain that my fellow Americans expect that on my induction into the Presidency

the specific term is really popularized by FDR's invocation of it starting in either 1938 or 1939 on his radio addresses (he alternates between that, My friends or my fellow americans and my friends). I'd chose september 3rd 1939 as the date (his fireside chat announcing the Nazi invasion of Poland)

From then on it becomes a staple of american political rhetoric.