The most likely reason is that the proverbs are easy to remember then.
Step-by-step explanation:
Alliteration makes things easy to remember as they flow off the mouth easily.
Proverbs are supposed to be made so that there is something in them that people instantly remember them.
This is why they use short and precise language, often very picturesque or revolving around a very singular idea or comparison.
This is accentuated by the focus of the alliteration which makes the sentences so as to make them be more memorable and more talk worthy.