Do they really 'demand a greater share' in proportion with how much more they contribute?

Police protect assets, fire services protect assets, schools are better in wealthy districts, The IRS targets poor and middle class because it's easier. Courts give better outcomes to the wealthy. Businesses are generally started by those with money. Pretty sure that the wealthy use significantly more resources because they have more to protect.

But where's "with how much more they contribute?" part of the answer? Such as the jobs created?

I would think that most public services, such as public schools or public transportation are consumed by lower and middle class people.

The upper class has their own private version of pretty much everything.

Even if true, you need to make that inference explicit, because that's a confusing, non-standard usage of "flat tax" and "regressive" tax.

Second, can you show your basis for thinking that? I would think it's the opposite: the wealthy aren't using the public schools, free health care, etc.

You know, that's probably debatable. The high wealthy also spend a lot on private services above and beyond what's publically available. Private security, planes, accountant and legal teams, private doctors, etc.

This seems counter intuitive to me but I’m not saying you’re wrong. Could you share more?