Every "self-regulatory organization" exists with the implicit threat that the government can and will impose external regulations if the organization doesn't do a good job. FINRA is authorized by law because lawmakers believe it does a good job of regulation, and that law continues to exist only because lawmakers continue to believe that. The SEC watches what FINRA does, and the government could easily choose to stop authorizing FINRA and assign its duties to the SEC.

In fact, FINRA itself was only established in 2007; the previous self-regulatory organization was the National Association of Securities Dealers, NASD - you may have heard of their Automated Quotation service, NASDAQ. FINRA was created after a long period of the SEC criticizing NASD for prioritizing the good of its business over effective self-regulation (see e.g. https://www.washingtonpost.com/archive/business/1994/11/15/s... and https://apnews.com/article/2d710b55afae651b8ad0737477ba2d79).