No.

A child is ineligible for certain public schools if they don’t have the corresponding residency permit, or Hukou. A Hukou is issued irrespective of whether or not you own your own residence.

The source of the confusion is that 90% of Chinese do own their own home. So the chances of you having the Hukou but not actually owning your own home is extremely slim. And vice versa if somebody is a renter, chances are they came to the host city for work, and thus no Hukou.

There are indeed people who purchase a property because they think of it as an investment. They won’t have the Hukou. Their children would be ineligible for various benefits that a local would have.

It is the Hukou, not the owning of a home, that determines eligibility.