Repair drywall and protect wires on back of electrical panel
I think your idea of a wood panel isn't bad, but I would go the extra mile here:


Buy a nicer board, a couple of basic hinges, and a knob
Paint the mounted opening white and fix your drywall
Cut your board to be large enough to cover the opening and paint it white
Attach the knob and hinges, then mount to the opening on the side closest to the wall


It looks like a cabinet, but if you slap a simple label of "Electrical" on it, nobody will attempt to use it to hang things, etc. A mere board may be confused for a bulkhead.