The cold hard fact is private uni grads will always be disadvantaged in the job market. And you should accept this fact, unless you possess clearly visible skillsets/achievements/personalities none of your peers/competitors have.

A smart HR/hiring manager will low ball you, simply because they can. No need to reiterate the reasons.
A good company/HR/hiring manager will of course offer you a fair salary, but you should already know there are not so many of such employers.

The stereotype of private uni grads = average/below-average hires will never change, unless the majority of private uni grad population can prove otherwise.

Stereotypes are formed by hearsays, personal experiences and certain biases. It's the same if you think certain nationalities are ruining a company's culture, or they should be paid lower than Singaporeans because of whatever reasons.

My personal belief is private unis can do more to make their grads stand out. Like we always hear or witness how SMU grads are more out-spoken. Many attributed this to their class participations. Why can't private unis do something similar?