Sorry to nitpick but...
The Doctor pushes a button which just... makes a random bit of platform disappear, killing the Sycorax leader. No attempt is made to explain this at all.
What was there to explain exactly? It happened in front of our very eyes. The Doctor just found the right button and he killed the Sycorax leader. Why should it be more complicated than that? Do you really think that the scene would have been better with a little dose of technobabble?
RTD had some good ideas but his obsession with fame, modern day London and the media means his work has dated very badly
That doesn't make a lot of sense to me. Fame, modern day London and the media all sound like contemporary themes to me, i.e. not dated at all. Or is Black Mirror
, which deals with those same themes, dated too?