I see no reason to presume that.
Not presume that, but
allow for that. That's the least we could do to alleviate the problems of the writers who take it unto themselves to create century upon century of consistency...
Except that Articles established that the mechanism you just described has only happened twice in UFP history: With Hiram Roth's death in the 2280s and Min Zife's resignation in 2279.
And that's KRAD's own damn fault.

He should have left the other incident hanging in the air, to allow for future stories. Who knows, perhaps some other writer would have turned that one into a major adventure for President Archer or something?
Seems to me that the early Federation would only undergo a truly dramatic shift in governance if it was politically unstable, but there's no evidence that any of the founding societies are such.
Unstability in a political assembly today doesn't require the individual representatives to be unstable... Just putting together the dissimilar if stable cultures of Vulcans, Andorians and Tellarites might have been enough to require three or four major reboots of the UFP system before it really got going.
In the end, it's not as if anybody has a competing candidate for the first UFP President. The old Franz Joseph manual suggests that a UFP Council Governor by the name Thorpe held the strings in the early stages of the game, signing the Romulan Peace Treaty; that, too, should be quite possible without contradicting
AotF as such. President Vanderbilt could have gained his powers after the signing, by means not used for electing later Presidents; he might also have served a term of nonstandard length.
Perhaps he wouldn't "count" in certain types of history writing because of that nonstandardness? Many a nation has started out with a leader whose powers were directly inherited by a series of successors, but who differed from those successors by virtue of being the first. Sometimes such trailblazers took on funny titles, but it wouldn't be impossible for the first person into the UFP Presidency to also refer to himself as President.
Also, a simultaneous President of Earth is always a possibility: Samuels could have had Littlejohn arguing politics with him off camera (except for Friedman's camera, that is). Especially if the relationship between the two were as ill-defined as in France. Or as ill-defined as the relationship in Finland has lately become, as the powers of the President are being stripped in favor of a more parliamentary setup.
Timo Saloniemi