The best way to go about with what-you-would want has to be done in 2 layers.
You need to measure not the size of the hole the button fits into but the rim of the push button.
One layer on the bottom which would seat your buttons, than probability a thin acrylic layer on top with the holes made just large enough so the rims of each button sits below or flush with the surface.
It is very very difficult even with the right tools to get a accurate bi-level cut in 1 piece of material. Multi-step drill bits have preset spacing, so they would not work with out making a huge gap. and using a router will produce an uneven surface on the bottom of that button hole causing improper setting of the buttons.
Also Almost no arcade panel I ever seen has recessed button rims. And you ask almost any Carpenter, cabinet maker, or any modder/ Builder on this or any other site and almost everyone makes their control panels in TWO LAYERS OF MATERIAL .