This is on the right track. You need a large area around the block to support the router base. You do not want to hand hold the workpiece for this!
You need to secure the block after you drop it into the opening which is made very precisely to fit. A threaded insert and rod the clamps the block in place ..... maybe two, one on each side at 90 degrees?
The top of the block must be flush with the larger surrounding area.
A rectangular frame/jig is attached to the "surround" to limit the router travel such that it creates the desired opening in the block.
Assuming your router base is round will make that easier. Two "L" shaped pieces can be adjusted to create the jig.
Once the block is secured inside the drop hole and the jig is screwed to the surround, you can begin with a shallow cut like 1/4" or so, moving the router inside the frame in the correct direction. This first cut will minimize tearout. Then you can make a second or third cut down to final depth.
You will not be hand holding the block which is totally unsafe. The drop in jig will secure the block well enough to permit the router to cut smoothly without vibration. This is critical. The jig is much like a milling vise on the vertical milling machine which can take very strong cutting forces in metal. The first cut it also critical for a clean result. Adjust the depth to get the best result.
Actually, a small vertical mill with a CNC control would eliminate the router altogether, but I don't think the RPMs would be fast enough for a clean cut in wood. A CNC router is the perfect method as suggested above.
Another approach is to bandsaw out the center opening by making an entrance cut for the blade and then gluing it back together. This would require a separate bottom also glued on afterwards.
If only you could "mold" wood in a press like plastic injection molding, this would be a perfect application.