When we specifically look at plot.background within the theme setting, we can set the parameters with element_rect(), providing us with the tools to change colour, fill, etc.
See: http://ggplot2.tidyverse.org/reference/element.html
However, I'm not sure how to set the parameters such that I can have rounded corners for the plot background
I've been discussing this with someone on SO after they saw your tweet! Others might have more useful advice, but we speculated that this would probably require a ggplot2 extension that modified element_rect to include a roundedness parameter and allow for the use of grid.roundrect instead of grid.rect where roundedness was given by a user (assuming a default of zero).
In other words, not presently possible but doable with work