1-1/8th inch is typical for regular pushbuttons... some of the weird lighted/shaped buttons from happ are like 1" ... and like you say, someone carved out some 1-1/4" holes (either not knowing they are 1-1/8th and used 1-1/4" or had a rough time with a 1-1/8th inch bit walking around and it ended up oversize.
in any case, how you want to go about this is going to depend on how you plan to finish the panel later.
if you are straight up painting it, you will likely end up having to weld around the edge of the holes to build them back up again and the use a die grinder/dremel on the holes to grind them to the proper size again.
if you want to start all over and/or move stuff all over and are going to have a pile of holes, get yourself a replacement panel. what you'll need is going to depend on the cabinet... but stuff like that is available...
https://twistedquarter.com/index.php?main_page=index&cPath=219_248if you are just covering it with plexi and artwork, don't worry too much about it. make sure your plexi has the proper size holes... the artwork will cover most of the errors in the metal anyway.
don't torque the buttons way down. just snug to keep them from moving during play... this isn't a machine out in public that people will be actively trying to pry and take ---steaming pile of meadow muffin--- off it at every moment.
alternatively see if you can find some 1-1/4 tube with a 1/16th wall (0.0625 wall), that will make a tube that is 1-1/4 on the outside, and 1-1/8 on the inside... you can cut some slivers off of it and create a shim to go between the button threaded portion and the panel... (effectively increasing the button shank diameter to 1-1/4". never tried this suggestion ... just throwing that out there.