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To be specific, control+v...that would seem to be the thing that defaults.  Instead, nothing happens. Should I just make a script that does this? Intuitively, as I mentioned, this 'feels' like it should work internally.

(+1)

I think you could try entering widget mode, then choose [Edit] -> [Paste as new canvas/ Paste into canvas] from the menu bar. Hope this helps!

The whole process: [Select tool] selecting part of card that you want to copy -> control+c -> switch to [widget mode] ->selecting [Edit] from menu bar-> [Paste as new canvas/ Paste into canvas]