On the basis that Bank is probably a convenient station for most people to
change to the DLR, it looks like a hell of a schlep to get to LCA.

Why? The direct service is 22 minutes (no changing).

I think I'll still avoid it, and use Gatwick or Heathrow.

You can get from Bank to Gatwick or Heathrow in less than 22 minutes? boggle

Even without the DLR connection the experience of using LCY is so much
more civilised than any of the other airports.