So I decided to build one myself, taking inspiration from a few videogames, where desks with built-in screens are commonplace:
As much as I like the look of the tables in the games, I decided to go with a more traditional look.
Step 1: Routing the ledge for the glass
The glass was centered on the table and marked up. I'd only be routing the ledge, I'd be cutting the middle part out entirely, to make room for the screen, so it didn't make sense to rout all that.
I also left the corners - once the middle was removed I'd use a forstner drill to do those, to ensure a nice round corner.
Step 2: Cutting out the middle
I did this by using a circular saw to plunge-cut on each side, using the routed ledge as a guide, then using a jigsaw to finish the cuts on each side, allowing the middle to drop out (which of course had to be controlled, leaving it unsupported would likely have caused the wood to split when the majority of support was gone).
With the middle removed, I could finish the corners of the ledge using the forstner drill.
The next steps are to build the legs and mount the TV.