There is no mechanism. Finance and startup Bros want to dump their overvalued companies with deceitful accounting on public investors to make a quick profit and socialize the inevitable losses. SOX makes it harder for them to do that.

To whatever extent SOX compliance makes it more complex to go public, it has no meaningful effect on legitimate successful companies - if you think going public will allow you to raise the most money, you'll go public; if ZIRP meant you could indefinitely raise private money, you'd stay private. Finance and startup bros want to blame companies staying private on SOX, but it has everything to do with either (1) the companies being deeply questionable from an accounting perspective, or (2) the companies being able to raise whatever investments they wanted in private without taking any of the costs of an IPO (e.g. the IPO pop, which could (simplisticaly) be thought of as money being made by the banks underwriting the IPO rather than by the existing owners/investors).