"But what about MTV - do they have a carte blanche, free for all, no holds barred, anything goes, no censorship and no boundaries on what they transmit - whatsoever??"
No, they don't.
However, you're not asking for censorship on the basis of explicit violence and sex. And by explicit, I mean "body parts that would get you arrested if you flashed them in M&S" and "violence that would get a 15 certificate". The former is fairly objective, the latter there's a fairly good understanding of (although certification debacles like Dark Knight show it's not quite clear-cut).
No, you're asking for censorship on the ideological content of the video. And you're asking for it on a three-minute by three-minute basis. The chances of getting any two people to agree on the criteria are close to zero. I don't allow shit like Eastenders in the house, because I think it has a low moral tone and shouldn't be broadcast pre-watershed, but that's my problem: I'm clearly in a minority. Others object to rap videos that demean women, to the satanism in Harry Potter or the fact that those nice David Attenborough documentaries talk about evolution. In all those cases, it's the parents' responsibility to enforce their mores, because it's simply not possible to get agreement on the classification of ideological content.
Film (and by extension TV and games) classification relies on an "exchange rate" between sex, violence and bad language which many disagree with (sweet romance Le Concert gets a 15 because one subtitle briefly includes the word motherfucker; had it contained instead a graphic scene of violence it would have got a 12).
Once you're outside a ticklist of immediate issues, there are a handful of "difficult issues" which will get you a certification issue, but Juno was (rightly) certified PG although a generation ago it would have got a AA or more, because abortion isn't as incendiary as it was. But although a PG, I suspect it won't get shown pre-watershed in the UK (I'd like to be proved wrong). I suspect many parents find that PG troubling: that's why there's the parents' BBFC extended classification information.
If people aren't aware that MTV show salacious videos, they must be living in the attic like Miss Haversham. If they think that there's any way to provide a monotonic scale of "badness" for pop videos and then assign times of day to each rating, they are sadly mistaken. If they don't think some of the content is suitable for their children, they have only one option. It's not as if I'm saying "don't watch the improving programmes on CBBC because they sometimes show porn", I'm saying "don't watch the mindless shit on MTV because there's other, worse, mindless shit on it as well". No one is going to be harmed, deprived or otherwise impacted by not watching MTV, and they might even read a book in the time freed up.