I can only assume that the credits were overlooked during production and they were contractually required (a Musician's Union agreement perhaps?) and it was cheaper to get the CA to do it than to re-edit the programme?
I think this was a regular thing on Play School, I've got an old Radio Times where someone writes in to say that the announcer always mentions who the pianist was but they never bother to announce who wrote the story. I'm guessing they wanted to limit the credits on the programme to an absolute minimum because they were of no interest to the target audience. Presumably it also saved time in editing and creating the credits if it was the same people all the time.
The Unseen Frank Skinner Show, one of those '90s 'too rude for TV' videos. It's from 1997 so only covers the first two series of the BBC1 incarnation of the show, rather than the better remembered ITV version.
I wouldn't say the ITV version was better remembered - the most memorable series was probably his last one on BBC1 in 1999, where it was pulling in huge audiences at 10.30, bigger than most primetime shows, and that was why ITV ended up poaching him. They had some big names on ITV, like Britney, but it was never the big critical and commercial hit it was on the Beeb.
I had this video, it's a bit bleak at points when the conversation gets a bit too adult. The stuff with Mick McManus and Mad Frankie Fraser* is a bit grim, neither seem especially pleased to be there. It's an interesting video, though, because it mostly came about because Series 2 was a bit controversial at the time because it was considered exceptionally rude for BBC1, and indeed the sleeve featured lots of newspaper cuttings slagging it off ("FRANKLY, WHO NEEDS GOOD TASTE"). And in Frank's book he talks about how there were a lot of last-minute edits as the Beeb got cold feet.
There's also a bit of an oddity with Mr Methane, the "stage farter", because I remember Frank mentioned him on the show and said he wasn't very good, and Mr Methane complained, so to make up for it they invited him on and Frank performed a duet with him. This never made it onto the show itself because it was too disgusting for BBC1, and it was mentioned on the sleeve of the video as being on it. But then there was a sticker on the box saying "Mr Methane does not appear on this video", presumably due to some sort of last-minute contractual dispute. In his book Frank said he really enjoyed it, and he says the funniest moment in his entire career is that, just before they started, Mr Methane did an incredibly loud, tuneless fart, as if he was tuning up. Mr Methane, there.
* Although the Mad Frankie Fraser interview is a bit dull, it did include Frank's favourite exchange, where Fraser was talking about Jack The Hat, and Frank said, in jest, "Just to make it clear, Frankie, Jack The Hat was actually a bloke, not just a hat?", and Fraser said, in all seriousness, "Oh yeah, he was a bloke, not a hat".