I am not sure I quite understand the vision that those arguing against greater press regulation have. It seems they think that the politicians themselves will be on the regulatory board, loftily able to gag any paper that dares to print anything remotely controversial about themselves.
This is not even remotely correct. Do you think that the ASA, for instance, is hopelessly corrupted by virtue of having been set up by statute and hence polticians? How about OFGEM, or OFWAT, or OFCOM? What about Doctors and their professional regulatory body, the GMC? This is the type of regulation that is being talked about.
And for all that new regulation might be brought in, papers/media ultimately have the public interest defence, which remains sacrosanct - If they can demonstrate that a story is in the public interest, no regulatory body in the world, no court, is going to stop them printing it.
If you are really concerned about politicians having undue control over the press, you really should be looking more into D notices and National Security Concerns, through which the government can suppress pretty much anything it wants, even now, when we have this "free" press.