While there maybe some truth to that, it's also worth noting that we see the same with all our (British) terrestrial TV networks. Particularly channel 4, which has given us classics such as The IT Crowd, Father Ted, The Inbetweeners and Black Books.
Edit: and Brass Eye. How could I forgot Brass Eye when this topic is specifically about satire? Awesome show.
I'll note that Channel 4 has a public service remit in its broadcast license, which, among other things, requires "demonstrating innovation, experiment and creativity".
Channel 5, otoh, does not, and we don't tend to get a whole lot of interesting new stuff from it - they like to play it very safe.
Let's regress this a level further - creating lots of small shows with potentially longer runs could result in a better talent pool (both actors/directors/producers), benefiting even private networks.
How often do BBC actors work with other networks? If there's any cross-pollination there is an argument that the BBC's reach goes beyond it's own productions.
Do not overlook the feeder stream here: alternative standup comedy in the small-club format, and the giant mixing pot of the Edinburgh Festival Fringe. There's a definite progression of free fringe show -> paid fringe show -> getting talent spotted and onto TV.
Since partial privatisation much of the BBC's output is actually commissioned through a maze of independents. It's not like the old days when Morcambe and Wise being poached to ITV was a national crisis.
It would be hard to dismiss BBC's influence on the format of British television since the BBC (channels 1 and 2) were originally the only TV channels on British TV. And, as you rightly pointed out, there's always going to be cross-pollination with TV talent as well. However I don't think it pays to look at the BBC as a unique model in the way that you are (ie "The BBC is different from American networks therefore it's the BBC that has uniquely shaped British TV"). I don't consider the fewer episodes per series/season model to be unique to Britain let alone a BBC invention. In America you tend to refer to these runs as "mini-series", but even them aside it's pretty common to have popular shows only run a dozen episodes a season.
I think the differences might be down to writing styles more than anything though. In the UK it's common to only have a 1 or 2 writers behind a popular series (or guest writers who handle individual episodes), where as in America you can have an entire team of writers. I would imagine having more writers might allow for a larger pool of ideas.
Edit: and Brass Eye. How could I forgot Brass Eye when this topic is specifically about satire? Awesome show.