All this really indicates is poor supervision of the event by its hosts. Seriously, this should be no more difficult as far as crowd control than a golf tournament.
To start with, there should be “divisions” by age, and “degree of difficulty”, of parts of the course. I’m sure this would be no real problem, and would help insure that even little children would have fun and competition with older kids would not crowd them out.
There would be referees on the course to handle egg disputes between children and settle down any children who are misbehaving. And parents would both have to remain in sight of their children and be quiet, and outside the search area.
Finally, once the children had collected their eggs, they would have to “check them out” before leaving, to make sure they were all “official” eggs. This could be done by putting a tiny RFID sticker on each one. As the eggs are checked out, any without a sticker, or that is visibly damaged, would be replaced with a similar egg.
Or you could simply stop following Ancient Sumerian Religious Rituals.