Unfortunately, even private companies have an ultimate obligation to maximize profit (or else they're overtaken by another business which will). This means that employers and employees ultimately remain opposed.
One reason for this is that it's hard to measure the value of an individual employee's contribution to the company [0], but it's easy to measure their cost. Since things that are easy to measure get weighted more highly [1], companies tend to see employees as costs rather than assets.
[0] except sometimes for salespeople who bring in big accounts, which is why salespeople in many companies are better compensated than engineers.
Nothing is in a vacuum, and employees (myself included) feel the opposite responsibility. I want a paycheck and to get that I need my employer to stay in business. There is a synergy there.
Maybe this is a scale issue. As a single owner of a business I feel personally responsible, and my actions in business are the same as in my personal life. But with a business I share ownership with, I have less personal responsibility, therefore by default I am simply not responsible (and feel less so) about the effects the business has.
Multiply this out to thousands of shareholders, where this connection of ownership -> responsibility towards employees dilutes to the point of non-existence.
[0] except sometimes for salespeople who bring in big accounts, which is why salespeople in many companies are better compensated than engineers.
[1] https://en.m.wikipedia.org/wiki/McNamara_fallacy