The consideration of these factors in the admissions process is determined less by the public or private status of an institution and more by its size and volume of applicants. Since schools like UMich and UNC Chapel Hill have low acceptance rates (below 30 percent), they're not just looking at test scores and grades and admitting students based on those numbers. They need to narrow the field further to cut down on acceptances. At any highly selective school, public or private, they're looking for students who fit both the academic profile and the personality of the school (or what the school needs at the time).
The assumption in your question does hold some validity, but it's more correlation than anything else. If a school is smaller, which is the case for many private colleges, it has to be especially discerning about admitting students that will contribute positively to the campus environment based on a holistic review. There are also fewer spots in the class in general. On the flip side, even highly selective public schools may focus mainly on academic qualifications for in-state applicants. Some states have laws about admitting in-state students that meet certain standards.
Your belief is loosely correct overall, but the reasons for the patterns you describe are complex. There is a huge degree of variation in admissions processes between schools, including those that share private or public status and have similar acceptance rates.
Here's a summary of my points:
In-state student applying to public college or university ----> less holistic
Out-of-state student applying to public college or university -----> more holistic
Any student applying to private college or university ------> more holistic
Any student applying to very large college or university ------> less holistic
Any student applying to very small college or university -----> more holistic
Any student applying to very selective (<30% admitted) college or university ----> more holistic
Any student applying to less selective (>50% admitted) college or university ----> less holistic
Take note of which qualities apply to your situation to decide whether your application will be considered on a more or less holistic basis.