The time pressure is probably more important than you realise.
The guests are often pretty eminent academics, feted in their field and used to being indulged. There have been some I know that very much enjoy the sound of their own voice as they tediously ramble for hours, bending any topic to their own pet themes, with colleagues and students obediently hanging on their words. Melvyn has the stature to get testy "Enough about his wife, you still haven't answered the question, get on with it!" and the Oxford emeritus professor complies.
The after show chat works because it's post-time-crunch. It's pressure release and reflection. If you do recruitment this is something to learn. You will have a much more valuable interaction after you have scraped off interviewee armour.
I generally agree with you and the "short" format is what makes it successful, but Melvyn said himself that they choose teaching professors because they would know how to explain subject clearly and after almost two decades of listening I'd say it has mostly worked.
> they choose teaching professors because they would know how to explain subject clearly
Hmm! The majority of academics in the UK teach... most of them reluctantly and badly because it's mandatory for their research contract!
Those that revel in it are used to monologuing extemporanously for hours every day in the lecture hall and supervision without interuption. It's quite far from a snappy conversational media performer.