Are you up to fastening connectors to each end of the wire? If so, almost any quick connect type would work -- anything from plain stereo 3.5mm plugs, to RJ-11 jacks. There's work involved in preparing each wire end, but once done, it'll probably hook/unhook easily.
If it were me, I'd just tape/glue/nail terminal screw strips on the underside, and screw the two sides in, like you already tried, though.
Another option, which is still used for some telephone installations I think, is wiring posts.
A third one is knife-based through-insulation clamping connections.