Seems like you are asking an honest question so I'll answer that question but only in regard to myself. I can't speak for other employers.
Early on in my business career, I did not provide health care. Of course, I paid the Medicare fees as required by law. I never had health care myself (until I was 55) and of course, health care used to be affordable, so I paid very high wages and bonuses without considering that cost. By the early 90s, I took on a partner, and he was appalled at the "excessive wages" but he was just as appalled at my failure to provide health care. So, we "made a deal" and I kept the high wages but we also got cheap health care HMO style and split it 50/50 with the employees.
I'm open minded enough to know that some businesses are low margin and employee cost can make or break them. I've never done business like that so I don't want to be unsympathetic to them. I've always hired a few less people and paid them more instead of more people, less wages.
So, probably you aree right in the sense that employers care only for their incomes and aren't up for sharing that income with their employees. From what I see lately, eliminating payroll taxes wouldn't go to the employee, it would go to the owners. So, no doubt it's my naivete speaking here.