But I can't reproduce what he says. When I press the sustain pedal at
the beginning of the tact all C#-s and G#-s get sustained and overlap.
That is what is supposed to happen. In the first edition, Beethoven's instructions at the start of the movement were "Adagio sostenuto", "Si deve suonare tutto questo pezzo delicatissimamente et senza Sordino", and "Semper pianissimo e senza Sordino" - i.e. he says "keep the sustain pedal down continuously for the whole movement" twice, but that doesn't work on modern pianos which sustain notes for much longer than pianos in Beethoven's time.
I suspect your hand position is wrong (as the other answers describe) because you are trying to hold down all the notes but your hands are too small. But the pedal holds the notes for you, so your fingers can let go.
The "4-5" fingering on the A in your score implies that the editor wants you to hold down the top notes, which is a good idea psychologically (even though it doesn't make any difference physically when the pedal is down) because it makes you focus on the melody part rather than all the notes in the triplet accompaniment. If your hands are too small to stretch the 7th C# to B with 2-5, play the C# with your thumb. The pedal will cover the "gap" when you move your thumb from B to C#. "Jumping" between notes with your thumb is much less important than not accidentally playing extra wrong notes like the G natural.