I did k-12 in the NYC Board of Education system (public school.) Some higher-end public high schools schools did randomized entry, which was a positive in my mind. The only selection was self selection into the lottery, where like-minded and ambitious students/families applied.
Unfortunately even that gets abused. I dont know how my process went, but I sure know how my kids' experience was. The school wont give you an application, or send you to the head office to apply (even though you can also apply in-school), or they will do the residency screening the last Friday of the application period (too bad if you happen not to be home on the day they visit.) They will sometimes ask you for original deeds or birth certificates (but your friends will tell you they werent asked for it.)
The randomness can be theatre to show public fairness, but in reality it is anything but random.