Why was the Hadith written 100 years after the Prophet?

That's one reason I believe Islam comes only from the Quran.

Why do the oldest manuscripts of the Quran go back to the Prophet's time, unlike the oldest manuscripts of the Hadith that only appear 200 years after the death of the Prophet?

I can't believe that scribes at the time of the Prophet didn't think it was a good idea to write down his sayings.