Because it is a Scherzo. In Beethoven's time, a Scherzo was expected to be a "light-hearted" and fast-moving humourous composition in the rounded-binary form. It had some characteristics, but three chief ones were as follows:
- It had to be short, and preferably a (the third) movement in a larger opus
- It had to be in 3/4 time
- It had to be fast, sometimes blindingly fast
And so he chose to go along with the convention and respect it. Beethoven has written some more Scherzi before this, starting with the 3rd movement of his Piano Sonata No. 2 in A Major, Op 2 No. 2 published in 1796. You can find the same characteristics in that. Or the 3rd movement of the Piano Sonata No. 3 in in C Major, Op 2 No. 3.
Of course, variations to this form began to appear later, and the rules began to be broken once in a while, but the form endured.
So, by convention, 3/4 was the right choice. However, Beethoven wanted more. The difference between 3/4 and 3/8 is on the emphasis. In 3/4 you have three quarter notes, and each quarter note is a beat, and the emphasis is on every one of them. In 3/8, you have three 8th notes instead, and the emphasis is on the first beat only.
His desire to respect and defy the convention at the same time however can be observed in the choice to embellish the first beat of every bar by adding that flute note at the top which makes it sound like a 3/8 composition, with the emphasis on the first beat. This was very novel at the time and can still thrill today, nearly 200 years later.
You can read more about the form here: