I would suggest you build with plywood, I have learned to hate MDF over the years, I do currently have a CAB that is 5/8" but still heavy.
As for 1/2" plywood your right that might look a bit off, there is 5/8" plywood but probably going to need to find a lumber yard as not typically something you would see in a big box store. Also keep in mind if building from that you probably want to use Formica on the sides of the cabinet, not only will this give you an extremely flat durable surface it will increase the thickness a tad.
If you were to just do the sides with plywood, and the front coin door panel with matching melamine (I'm assuming you would do black) and that in itself would take off a significant amount of weight.
As for the Casters at the bottom just make sure you use something like 2x6's incorporated to the frame of the Cabinet and you will have plenty of material to mount those with (see a photo from a cabinet I'm currently working on, 4 casters on the bottom) If you do use casters make the front two swivel and the back two fixed.
As for your thoughts on MDF being "Classic" that 3/4 stuff was primarily in my own antidotal experience on Atari Cabs, Several other manufactures used plywood, Hell Nintendo has an extremely strange width of plywood with the Formica added = 9/16" thick. I would suspect if you have a good Bezel that is 2" in from either side you won't really notice the width of the boards anyways.
EDIT: one last tip, make sure when you make it the width is small enough to fit through a standard door frame, i know it sounds elementary but you might be surprised. Especially with control panels! If the CP does come out more than the standard width (what 36"? ) Make sure it can easily be detached for when moving the Cab