An airworthiness certificate is not required for individual parts like bearings. The aircraft in question, say for example the Boeing 737, receives an airworthiness certificate for the entire plane. Any parts that are brought into the assembly process before or after certification (which the Boeing 737 already has) only need an SAE approval to qualify as a direct replacement part. The FAA does not certify parts. They approve of parts that meet certain standards, one of them being a listing by SAE. If you decide to replace an entire system in the aircraft with an updated substantially different one, as with avionics systems upgrades, then the FAA has to certify that system for a specific aircraft. But they don't look at each part of the new avionics system and approve or disapprove it. The FAA is a government regulatory agency. They're not engineers. They have neither the time, resources or testing labs to do that. They rely on parts that have already gone through a certification process by SAE. And SAE doesn't even have the resources to do all that. They require the manufacturer of any part that wants approval to go through an approved test lab.
That's why there are SAE standards like SAE-AS81820 and SAE-AS81934. If your part can meet that standard as verified by an official test lab, it gets SAE approval and can then be used in an aircraft without it's own airwortiness certificate.
FYI from the FAA.gov website:
Sep 30, 2015 - "An airworthiness certificate is an FAA document which grants authorization to operate an aircraft in flight. Who may apply for an airworthiness certificate? A registered owner or owner's agent of an aircraft may apply for an airworthiness certificate."
