ZCI isn't going to do any good in the center areas where the grooves and dadoes intersect. The best solution that comes to mind to prevent the tearout altogether would be to cut some strips of MDF that are the same size as your dadoes/grooves. Cut the dadoes on one side and carpet tape mdf strips in. The cut the grooves on the same side. Carpet tape strips in. Flip board and repeat. Sounds like a PITA but if you precut the strips to be a tight fit, it's less of a pain than sanding

). Using this method would leave all of your cuts backed up with sacrificial material and should greatly reduce if not eliminate the issue.
The other option that springs to mind would be to make scoring cuts along the edges of where each groove/dado will go then go back with the dado stack to finish it. As long as your scoring blade is sharp and close to the final depth of the dados it should be effective in improving the cut quality at the edges.
I suppose this goes without saying but using a quality dado stack that is clean and sharp is the most important step to improving the cut quality…
Finally, I really like that trivet design! Very elegant but also a simple design to execute. That's the kinda stuff that makes people money ;-) I wouldn't be surprised if something similar ends up under my Christmas tree for someone.