After exporting from the original site and importing into the new site, all widgets were successfully imported and showed up in the website, but the widget admin screen was rendered unusable. All the "dynamic" elements such as clicking on sidebars and widgets to expand & collapse, or hovering over the left menu to reveal options, did not work anymore. I could do nothing in that screen anymore.
I have two plugins that do stuff in the widget admin screen: Polylang (adds a language selector to each widget) and Slim Jetpack (adds a visibility selector to each widget).
I couldn't find any way to make it work, so I had to delete the widgets directly in the database.