I don't understand how either of your proposed angles of reform are supposed to work.
These are services being paid by the state, so I would think the state already has direct say in pricing? Which is why the business becomes a march to the bottom in provided services, to increase margins by decreasing expenses since they can't increase prices.
I didn't see any mention of bankruptcy in the article, so I don't see what limited liability has to do here. If anything it seems like the entities are too well capitalized, in that they can fight with individual states and pay off whatever meager fines might result. Also as far as I understand the medical industry, the individual doctors/nurses signing off on the care being given are still directly liable (modulo professional insurance policies). They just have reams of paperwork saying the care was correct from their perspective, despite what the low level caregivers might be doing outside of the paperwork.
I personally think what limited liability entities have developed into has become a scourge in general, as they end up being considered to have all the rights and freedoms of personally-liable natural persons. Getting a government granted liability shield should mean submitting to significantly more regulation to preclude engaging in well known patterns of constructive negligence. The vaunted fiercely-independent "man in the arena" can always actually get into the arena if they want their business to partake of the full natural rights afforded to natural persons...
And maybe you're just coming from a similar feeling of wanting those ultimately in charge to also be directly responsible? I usually think of this in terms of corpos fighting regulation with specious justifications based on individual rights (Citizens United, corpo direction of employee speech, etc). I just don't see how adding the liability would make anything more actionable here.