negligible91hayek_novice, I think it would be good to define some terms first to make what Dr. Woods was saying more clear.
Rothbard made a distinction between the incidence of a tax and the indirect effects of a tax.
The incidence is who the primary effects of the tax actually fall on. The indirect effects are the effects on individuals who trade with the individual whom the incidence falls on.
To clarify, Rothbard states “Thus, if an income tax is levied on Jones at 80%, this will hurt not only Jones, but also — by decreasing Jones’ incentives as well as capacities — other consumers by reducing Jones’ work and savings. It is therefore true that the effects of taxation diffuse outward from the center of the target.”
So the incidence of a payroll tax partially levied on an employer will fully shift toward the employee. That’s what Dr. Woods means. The employer will still be hurt through his interaction with the employee. If the employee was better off, then so would the employer.