Of greater annoyance to me is that BBC Four have repeated the error of last week by again showing the latest edition pillarboxed on the SD channel!
If you are watching on Freeview then all 4:3 content is pillarboxed on BBC Four SD "in the background" as all the BBC SD channels broadcast permanently in 16:9 (or certainly used to), and use AFDs to trigger receivers into 4:3 mode If an AFD flagging 12P16 is sent with the 12P16 content. There is no resolution difference in this case.
I believe the same is true of DSat SD - except the AFD is used to trigger an ARC
in the Playout chain (and the MPEG2 encoder is switched into 12F12 mode). I don't think that the move to HD Playout has changed this - so I again I don't think there is a resolution improvement for 12F12 broadcasts - though the MPEG2 blocking artefacts will be a different shape I guess, and there might be a marginal improvement.
If you are watching on a 4:3 TV configured for letterbox - I can understand your complaint though.
All of the above is based on the situation a few years ago - don't think it's changed - though I guess the DSat feed could benefit from additional resolution in 4:3 mode IF TOTP is now cached 4:3 SD 12F12 or 16:9 HD 12P16 AND the 12P16 to 12F12 ARC in the DSat chain is actually a 1080i downscaler rather than taking a 16:9 SD downscaled input...