Half lap joints with 2x4s is beyond overkill for an arcade cab.
Instead, get some 1x2 material. Better yet, if you have access to a table saw, rip your own 2inch strips of 3/4 inch plywood and butt joint them with pocket hole screws (and glue). This will produce a VERY strong carcass to build you cabinet around.
Metal brackets will also work, but I find them harder to use.