The distinction between employer and employee taxes is irrelevant, they're both taxes on labour.
They are both taxes on labour, you're correct. That is the view looking at it from an employer's perspective though.
A consultant on net pay of X costs the NHS and the HSE almost exactly the same amount.
But you did your calculation in the context of the pay scale need to attract someone to a job.
If I'm an employee looking at a job offer in two different jurisdictions, I'll do a quick calculation based on the gross pay figure offered to me, and the current tax regime and see what my net pay will be in each. That'll inform my decision greatly.
Suppose then the UK signals that employer's NI is to be doubled, and at the same time in ROI, income tax is to jump by 10%. In the UK scenario, I'm not affected (obviously my potential employer is furious), but the bottom line for me essentially doesn't change. UK isn't any less attractive as an employment destination, while the ROI tax change would affect my take home pay significantly.
I'd be surprised if there's more than 1 in 100 employees that even consider employer's PRSI as a factor in their pay. It's actually a fairly insidious tax as most employees just blithely assume that the tax they pay on their job is simply the difference between gross pay and net pay.