If you want to maintain the layer structure then try this:
Choose a new background colour by clicking on the background colour square in the toolbar
Add a new layer using the shortcut Shift+Ctrl+N, hit Enter to accept
Do Layer > New > Background from Layer
That's just 3 steps, one less than yours.
For more speed, you could record steps 2 and 3 above as an Action. And set up a Function key to run the Action.
Then when you want to use it, all you'd need to do is select a new background colour, and hit the Ctrl+Function key to run the Action. That's just two steps!!!
For the ultimate speed, if you want to apply this to a whole bunch of images in one folder, you could run the Action using File > Automate > Batch - choose the Action you recorded, set the source folder and a destination folder, and OK to run it.