If you still want to enable this ctrl-f to enable the site wide search, I suggest making it opt in with the checkbox. It's really obnoxious to have to disable the checkbox.
This search interface has been a huge hit with our regular users, but it's harder to get feedback from fly-by traffic, so this is really helpful.
To be honest, the non-standard behaviour plus strange looking popup confused me. I didn't even notice the tiny checkbox, nor that you could press CTRL+F again to get the standard search behaviour. My mistake, of course, but this speaks of poor UX in my opinion...
The user could get the option to prevent (by default) changes to bindings, with the option to permit them on a site-by-site (or even page-by-page) basis; or to allow them by default and then deny them on a site-by-site basis.
The downside is that the site's built-in search feature is always worse than ctrl-f at finding something on the page. Always. It shouldn't even be hard, but they're always returning garbage like 8 year old posts on completely unrelated articles instead of the stuff on the page you're currently viewing, and advanced mode is usually broken in weird and confusing ways. It seems like web designers are optimizing for the thing you never do.
I just switched to reader mode and used Command-F there instead.