Here is a little history on SFI. It originally was founded as an arm of SEMA which is the Specialty Equipment Market Association, a trade association. SFI originally stood for SEMA Foundation Inc. SFI does not actually "approve" anything. They set standards for a number of products that could be potentially dangerous if they were not correctly manufactured. These standards originally covered wheels but then safety equipment, dragster chassis, harmonic dampers, and many other products were added.
In the early eighties the SEMA board of directors became concerned about the potential liability of SFI. If someone got killed or injured due to a part that had been certified as meeting the SFI standards, SFI could be open to a lawsuit. So SEMA spun SFI off as a separate entity. At that time, SFI did not have enough income to support the staff so SEMA continued to make cash donations to the organization until it could get on its feet and support itself which it has done for a number of years.
SFI itself does not test anything. They set standards and then the manufacturer must undertake independent tests of his product to assure that it does meet the SFI standards. If any manufacturer advertises that his product is approved by SFI he is violating the SFI regulations. The correct terminology is "This product meets SFI Specification #18-1." Or whatever number spec it has to meet. Each product sold as meeting an SFI spec must have an SFI label on it. The manufacturer must buy these labels from SFI which is how the organization supports itself.
Most racing organizations require that you have SFI spec parts. In many cases you are better off to pay the extra money and get an SFI part. In the case of harmonic dampers, SFI spec dampers must have a provision for retaining the inertia ring in the event the damper fails. This means the ring can't fly off and go through your radiator or tear up some other part of your car.