Doctor Who hasn't really dipped its toe into satire that often. There's The Curse of Peladon, an allegory exploring the issue of whether Britain should join the EU, The Sun Makers might be a comment on increased taxation, Vengeance on Varos came about during the video nasty scare, and of course The Happiness Patrol was a big slap against Margaret Thatcher that was so smug and pleased with itself about it.I myself found the Slitheen arc to be rather dumb. But I used that as an example to refer back when Doctor Who had a mildly satirical nature, and was made rather cheesily. Now the series itself is way too serious.
I'd be hesitant before including The Long Game and other RTD-penned episodes about reality TV as "satire", though. There's no insightful commentary, no real criticism, it's just basically modern reality TV but made more extreme. Over 200,000 years, you're telling me Big Brother, The Weakest Link and What Not to Wear will survive in the exact same format with robot versions of long-dead presenters? They're there because a modern audience will recognise it. RTD was concerned with keeping the show up to date, and felt that introducing modern issues and topics were the way to go, with a bit of tongue-in-cheek humour.
I don't think you'd ever call the Slitheen "satirical". Unless they were a very sophisticated parody of godawful monster designs and effects that just went over everybody's heads.
The best Doctor, as anyone knows, is Jon Pertwee. Now there was a Doctor who dressed like a don, was not afraid to smack people about, and got things done, man. The getup they've given Capaldi reminds me of the Third's dress sense, so if this means we get to see Malcolm Tucker travelling across space punching aliens in the face, then Moffat is the best thing to ever happen to the show.