This is because there is a Ctrl+Shift+L shortcut for the Clear Terminal Buffer command (left over from before I made Ctrl+L clear the Terminal instead of the Console depending on focus).
So as soon as there's a terminal instance, Ctrl+Shift+L starts being routed to it, even when it isn't visible.
I think I should remove this shortcut altogether, since Ctrl+L already does the trick.
In the meantime, users can modify the keyboard shortcut and remove it themselves. Tools / Modify Keyboard Shortcuts, type clear terminal in the filter, double-click on the Shift + Ctrl + L, delete it, and Apply.