Is there a way to “hide” Vim’s interface to see the underlying shell?

Try Ctrl-Z, also accessible from the built in command :suspend. Does that do what you want?

If you do not mind not having scrolling, use

:!

(That is, run the ! Ex command without arguments.)

Otherwise, if you are running the terminal version of Vim, use :suspend or Ctrl+Z (the latter keyboard shortcut works in Normal and Visual modes).