Barrister/Solicitor in UK in "Broadchurch"

Also with the prosecution apparently all being conducted by a private individual and at the behest of the victim's family with no apparent involvement with the state at all. That just seems really weird to me as an American.

FWIW, private prosecutions are definitely a thing that exist in the UK. Unfortunately they don't track numbers so I (or, I think, anyone) can't tell you how common it is. http://www.cps.gov.uk/legal/p_to_r/private_prosecutions/ is the Crown's fact sheet on it. Not a UK lawyer and haven't seen it so I have no opinion on the main thrust of your question.

I'm not a UK lawyer but I study the UK legal system and my sense is that it's very unlikely that a prosecution for murder or manslaughter would be privately conducted rather than conducted by the CPS. That seems like the definition of a standard case where it's in the public interest to prosecute and the idea of a private prosecution is especially weird given that the case was preceded by a police investigation. I haven't watched the episode but is there any explanation of why the police didn't refer the matter to the CPS once they decided there was enough evidence to prosecute? The only reason I can think of is that the police and CPS decided that there wasn't enough admissible evidence to secure a conviction and abandoned the case and then the family decided to try a private prosecution as an alternative.

He pleaded not guilty at whatever the equivalent to an arraignment hearing is, immediately after which the worried and distraught family of the victims, on their own and on the advice of someone I don't recognize, decided to approach this supposedly highly regarded lawyer to prosecute the case. For fear, I guess, that a CPS prosecution would be ineffective. I'll take a look at the scene again... Okay, the quote from that person after being asked "what happens now?" was "It's a full trial, I'm afraid. We need to make sure that the CPS finds you the best QC..." What does QC stand for? Can you answer my question about the solicitor/barrister distinction, too? Is it that in fact this distinction has mostly broken down in at least these kinds of cases or is it that the show has just conveniently ignored the distinction for dramatic purposes? (And I also sort of wonder how influenced by American media courtroom dramas this all is.)

QC = https://en.wikipedia.org/wiki/Queen%27s_Counsel. CPS can engage private barristers to assist them in complex cases, says Wikipedia. Haven't been watching Broadchurch myself.