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 .
For recessed buttons that I have done was I marked a pilot hole with a 1/16th drill bit on a drill press. I then proceeded to take my 30mm forstner bit and create the recession for the button. Nextly I took my 24mm forstner and drilled for the 24mm button to go through. There will be just 2 millimeters of space approximately around the 24mm button in the 30mm hole though but that’s okay.
The other method would be to mark out your circle and use a flush router bit to make the recession around the buttons
This is called counterboring. Unless you’re getting fancy, you can probably get away with cutting the wide hole with a spade bit and then drilling out the deper one using the original hole as a pilot. If you care a lot about the counter bored surface you’d have to use something more like router bit or end cutting mill (there are, of course, many other possibilties like dedicated counterbore tools, but that’s probably massive overkill).