Plywood is stronger, but it warps and delaminates when wet, and unless you're talking marine grade plywood it has voids in it. Marine grade plywood is significantly more expensive than MDF.
As to MDF versus particle board, MDF cuts and routes nicer, and needs minimal surface prep to finish. The cabinet making industry switched a long time ago to MDF for that reason.
The arcade cab manufacturers of the golden era were stuck with what they could get to build the cabs. Demand for the cabs was huge, and sourcing the materials for a several thousand cab production run ment they had to make do with what was available in sufficient quantities.