I think the qualm some people here on TUG have with the BBB is that the BBB accredits these TS scam companies in the first place. From what I understand (I could of course be wrong), a company can just pay the BBB a membership and start out with an A+ rating and the BBB is more than happy to accept the membership fee without vetting the company. I remember a number of years ago I had an upfront fee company call me and she started out by saying, "We have an A+ rating with the BBB." So either she was lying (which is entirely within the realm of possibility) or the BBB is accrediting a scam company.
The other qualm some here have is that if someone gets scammed by a BBB-accredited company, the only recourse that person has is to file a complaint with the BBB. All that filing does is give the company a lower rating. Then, after the company has been called out, it just shuts down and resurfaces under a new name and recycles the process. The BBB has no power to give the victim restitution or to have the scammer charged. All the victim really accomplishes is venting.
You can not "buy" your BBB rating. To qualify for Accreditation, you need to have a rating of "B" or higher. If you were to start Lanny's Shop tomorrow, you would start with an A- rating, which is the baseline for all businesses. You would need to be in business for at least a year in order to rise to an "A+" rating. You could not become Accredited until you have been in business for at least six months. Let's say you sought Accreditation and were approved in month seven of operation. You wouldn't be eligible for that "A+" until you had a year in business.
You are correct that BBB has no power in ordering restitution on any complaint. The law enforcement arm in each state lies with the Attorney General's Office. We remind consumers of this and can give them information on how to file with the AG's Office. However, we often get resolutions where consumers get their money back.
As for the businesses that perform poorly under one business name and open under another, our office in St. Louis is aggressive in producing consumer warnings. In those warnings, we not only name the business, but also the business owner's name. We believe it is important that the owner's name is out there in case he/she decides to do what you have described. Our consumer warnings have a shelf life of 36 months online, which is as long as our complaints and customer reviews live until expiration. Our hope is that people do research on the company and its owners before doing business with them. Unfortunately, we mainly hear from consumers who may not have done all of the due diligence necessary prior to their purchase.
If you have any more questions about BBB, I would be happy to answer them.