Weak does not mean fabricated. Weak just means that the hadith sciences can't be used to demonstrate it's authenticity to any significantthe highest degree. It honestly says more about the stringency of the verification process than anything about the quality of the hadith itself.
The sunnah of the prophet is evidenced through his words and actions throughout his entire life, not just those few moments that were perfectly recorded and passed down by verifiable channels; there were decades between the first revelation and his death, and every Sahih hadith ever written barely scratches the surface of that time. Many who care about following the sunnah are loathe to lose any of that wisdom he demonstrated, even if it means taking a bit more effort to understand and work out what's actually true through the weaker chains.