From reading the article, what the HSE are doing in many respects is not that dissimiler from what many large private companies do. Staff get sick, that's a fact, and on occassions, staff have long terms sick issues and given the size of the HSE, the numbers involved may not be excessive. It would be quite common for companies to pay sick pay for 6 months and then half pay for another 6 months (every major company I've worked for over the last 20 years has had similer policies)
However, it is after the 12 months that the HSE seem to be slipping up. Many large companies at this stage would have a private medical insurance scheme which would kick in and would pay a proportion of the employees wages, in the case of my current employer, they pay 66% of the wages after 12 months, employer pays nothing. Under the terms of the scheme, that can continue indefinately (assuming there is no change in the underlying condition).
Even where companies do not have such a scheme, my understanding has always been that an employer has no obligation to hold a job open indefinately for someone who cannot physically do it. Therefore, and depending on the circumstance, either resignation or redundancy would kick it
To me this has nothing to do with the merits or demerits of privitisation of the health sector. It's just crap HR management and policies by the HSE. That happens in plenty of private sector companies as well, but if the willingness is there to fix it, it's not a difficult thing to do