The root of the question comes from the incorrect assumption that in Beethoven's time (and earlier) the notation for dotted rhythms was performed strictly according to the math. The math was certainly "strict" in the sense of showing the mathematically correct number of beats in the bar, but that was not necessarily how they were played.
A single-dotted note could represent anything from modern "light swing" (i.e. less than 2/3 of a beat followed by more than 1/3, but not an equal division of 1/2 + 1/2) up to even 15/16 followed by 1/16 in a slow tempo (i.e. a triple-dotted note, in modern notation)
There are two practical options for playing these particular dotted notes: either 2/3 + 1/3, or 5/6 + 1/6. The choice really depends on the tempo for the whole movement - if it is too low, 2/3 + 1/3 sounds a bit lame and "dragging", and if it is too fast, 5/6 + 1/6 sounds like a little "click" at the end of the beat rather than something musically significant.
Of course if you are playing using rubato, the beat division doesn't have to be "mathematical" anyway - just play it the however you want it to sound.
Composers and music copyists didn't like writing "triplets" consisting of a quarter note plus a half note, because it meant that some sort of bracket or slur was essential to show that the notation was a triplet. when some of the notes in the triplet did not have beams. A dotted eight plus a 16th, beamed together, was simpler to write, and everybody at the time understood what it meant.