You can't add games to your lists on mobile. You can click the button, but the list selector that pops up has a transparent black overlay on it and clicking anywhere dismisses the list selector.
You can replicate this on desktop by resizing your browser window to be narrow enough for the site to relayout to have the navigation bar on top.
Since the mobile site is reactive to display width instead of a dedicated mobile site, I can't workaround this with "request desktop site" either.
I tested with Chrome on Android and on macOS, but I've encountered this problem for the past year.