Hmmm. I personally like the glass to be inset more than usual. So 3/4" isn't too "deep" into the cab for my tastes. A lot of people try to match this inset to the other inset of the cab, the sides, top, etc. The real reason for doing this (other than laying the monitor flat like a cocktail cab) is to cut down on the glare from other light sources. Remember arcade cabs were typically lined up one next to each other, so having the monitor pushed deeper into the cab gives the gamer less glare issues and more "privacy"? Whether you do this bottom router job on a scrap piece glued to the admin panel, or cutting the admin panel itself is entire aesthetics. Do you want the admin panel to look the same thickness as the sides? Since this edge is not a structural piece, you probably don't need screws. Just put your routed scrap on there with some glue, then clamp it LEVEL. Trying to make this piece completely square to your sides will help avoid issues later on when you have the glass cut. Speaking of the glass, I would say a few things:
1) Not all openings are completely square or at 90 degree angles. Measure each dimension in 3-4 places and make sure you either take the lowest measurement or have the glass people cut a non-square piece.
2) Talk to your glass person about their precision. Can they cut within 1/16", 1/18" etc. Only by knowing their precision can you make adjustments to your numbers.
3) Expansion. This probably isn't too big of an issue, but if you make it SUPER tight, be aware on hot days or when the monitor heats the glass it will expand and can break. I think since the size is relatively small, 1/16" on each side sounds okay. However if your glass guy is +/- 1/16" then this might not be okay. I think you get what I am saying.
Another option for the glass is to route into your side panel the length of the glass. Then open up your top so you can slide the glass all the way down to a 3 sided channel. This removes the gap that would typically be there and the thickness of the dado can be backfilled tighter if laminating the surface. This is a much more challenging setup and needs to be thought about in the design while building. I don't know if your cab allows for that, however, you could have 1/4" dado on each side and have the glass cut with a looser spec and still not have gaps on the sides. just something to think about.
If you're already committed to just laying the glass, pushing up, then setting in the bottom channel, I would just leave those gaps there. The glass need to expand/contract. Whatever you do, don't put U-channel on the sides to "tighten up" the glass, if it needs to expand from the monitor's heat, it won't be able to.