Where does it say that type certificates are required?
I guess I was thinking about it the wrong way. I was hoping for a single regulation to cover it. These sections of the FAR seem to be enough to say that type certificates are required for aeronautical products.

14 CFR §21.45 states the privileges bestowed to type certificate holders including the ability to obtain airworthiness certificates for aircraft, install engines and propellers, obtain production approval or replacement part approval.

14 CFR §21.6 states the need for a type certificate in order to manufacture new aeronautical products

14 CFR §21.9 states replacement and modification parts need to be produced under type certificate

14 CFR §21.132 states that a type certificate is necessary in order to be eligible for a production certificate.

14 CFR §21.183 states that in order to get an airworthiness certificate the aircraft must conform to its type design.

There are probably more examples in other sections. Any other answers appreciated.