There might be another option.
There's some coating that gets placed on eyeglasses lately to completely (or nearlY) get rid of glare. Have a look for the stuff. The pricetag might be steep, but you won't know without looking.
It's on Amazon, or at least something similar. The brand they sell is "Glare Buster" but I'm positive it's sold under a different name. My father tried it once, he didn't like it at all. I guess the principle is pretty basic and that's the drawback. The stuff adds a coating to the glass, essentially dull clear coat (as in paint). Apparently it works fine, but it tends to blur things just a tad bit. For someone wearing glasses, that's unnacceptable. For an arcade machine, maybe it would make it more authentic?

Laptop screens have a kind of anti-glare coating that doesn't blur the image. Not sure where anyone can get a clear laptop-like piece of material big enough to cover a cabinet glass. Did a little bit of digging and there's something called
Duralan II but that ---Cleveland steamer--- looks like it's only available to wholesalers and at custom sizes

Maybe try junk electronic stores to see if they have any in stock. Or find a glass shop that can coat the glass for you. When I worked at Home Depot, anti-glare coatings are options on most windows and shower doors (why shower doors need anti-glare coatings is beyond me), maybe a window shop might have a solution.