Consider: if a company had no profits at all, then for some period of time it would still (1) pay its employees (2) pay the payroll tax on behalf of those employees. The amount of payroll tax is 100% determined by employee compensation, not corporate profit (even if those two are indirectly connected).

You're more or less arguing that employee compensation is a tax too, since it also "raises the price the company has to pay and thus lowers profits".

Just like any other tax on employee income does, yes; unlike taxes on company profits, and like other taxes on employee income, it's effect on the company can be minimized by finding (or developing) and adopting less labor-cost intensive business methods even if they are higher (before tax) total cost.