I was talking to someone who's been involved in elite university admissions, and gaming for the city/neighborhood/HS has been a known tactic for a while.

There's a related tactic for people in affluent suburbs, for getting their child into urban school districts with an especially well-regarded primary public school: buy a house or condo in the city, and try to make it look on your application like that's your primary residence.

At that point why not just send your kids to a private school?

Depending on where you live, the break-even point of doing this versus a private school can be under 2 kids (and is usually under 3 kids), so you save quite a bit of money.

Many do, but some public schools are very well-regarded by parents, such as: https://en.wikipedia.org/wiki/Boston_Latin_School

Money. Private schools can be $40K or more (after taxes) per child per year. You can get a lot of real estate for that money instead in a lot of markets.