Why would private regulators be more efficient? Regulation will always seem inefficient no matter the nature of the regulating agency because it has different goals (ensuring safety and efficacy) vs making a profit for shareholders.

Public regulators are always preferable, when a market needs to be regulated as you admit, because the public, through the press and their elected representatives, can hold them accountable for their actions. If someone from the Mayo Clinic is receiving improper benefits from pharmaceutical companies how does the public find that out? There are no laws that would force disclosure of those relationships.

In the case of public regulators like the FDA, Congress can force its leaders to testify and the press has tools like FOIA to get at private documents. They're not perfect and can still lead to corruption and regulatory capture, but are preferable to private organizations with no public accountability regulating the pharmaceutical industry.