More on hospital organization (modular), and health care insurance
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My step sister (V) mentioned that it worked quite well to have a
larger hospital operate as a collection of multiple independent
organizations. That appears to be a useful idea. Semi-independent
(or fully independent) smaller organizations are much easier to manage
themselves, and they will create a market / comparison / competition
(even if merely in culture/theory) between them. Something good,
a good example, can spread, a bad example is more easily corrected;
the separate horizontally equal managements can use each other for
inspiration and be corrected. The same mechanisms which operate
between sovereign countries.

These separate organizations within the same larger hospital (employing
thousands of people) would do their own accounting and money flow,
their own people hiring and firing. The units could for instance
focus on a certain aspect of medicine, and do all the necessary work
around it, from cleaning to maintenance.  An alternative would be
to separate per profession, however this would undercut the ability
to compare since it is hard to compare brain surgery with cleaning
and so on. But if most organizations have "medical specialists" and
"cleaners" and so on, things could be compared.

Apparently this worked very well at the location where she experienced
it, it could be something to replicate further.

The smaller sub-organizations could also make the whole hospital more
transparent for both the state, the public and the patients. 

Elsewhere on this site was suggested to make the hospital infrastructure
state owned, because usually bankruptcy is not an option with a large
hospital; but one could always fire everyone and keep the infrastructure,
hiring new staff in case of a "bankruptcy." That would be a benefit of
state owned hospital infrastructure, it remains outside of a bankruptcy,
the hospital - if positioned in an independent way in the free market -
could rent the infrastructure and the more expensive machines from the
state. Such a set up could make a bankruptcy less costly: fire everyone,
create a new hospital group/company, and re-hire everyone that isn't to
blame for the bankruptcy and/or is of good quality (and so on). If it
can be assured no new bankruptcy occurs, one could even re-hire everyone
immediately. Some sort of danger with that is that doing business with
a hospital, maybe some other businesses would feel threatened by the
idea that the hospital could declare bankruptcy as a means to destroy
their payment obligations to them. (Maybe a little far fetched danger.)
But if the state owns the infrastructure and larger machines or even
all inventory of the hospital down to the last cup of coffee, then the
state would be doing business with other industries (the hospital would
want to buy something, the state buys it and rents it to the hospital:
the state is liable to the machine producing companies to pay the price
of the machinery.)

By cutting a large hospital up into smaller units, in theory perhaps
one could reach the size where it can be run like a normal for profit
business, who gets its revenue from the state health insurance bureaucracy.
The "useful market pressures" effect being mostly the ability of patients
to choose their own hospital, and the state deciding to reward moneys to
the hospitals according to their successes and general good behavior
(assuming that the state is truly the will of the public.)

It may still be a good idea to run these hospital units as democratized
units, either only internally democratized and externally depending on
the state health insurance to pay the costs and renting contracts for
infrastructure; or also enveloped into the management structure elements
of the patients and/or the state (directly). A problem with more
"independent" units is that they can have a monopolized business: there
being only one "heart surgery" unit, one "birth" unit, and so on. This
is not enough competition within that single hospital to speak of true
free market cleaning effects. Such effects would only occur between
different hospitals and their respective "heart surgery" departments,
"birth" units, and so on. Despite it being a strange kind of business,
a semi-public business or highly Government dependent business, the idea
of managerially independent units is probably still functional. It would
probably function, even if it isn't quite a free business like others
(who would own their own properties and machines, and lose them in case
of bankruptcy, and do business with free market actors, people and the
state alike - the hospital unit would not own its property, not own its
machines and perhaps own nothing, not lose these in case of bankruptcy,
only do business with the probably nationalized (mostly) health insurance
on behalf of patients who do have the free choice to go to that particular
hospital). Maybe there isn't a good and simple word for it, or an obvious
and clear mechanism (either nationalized like roads, or free market and
free initiative), it would probably be enough of a simple structure that
it would work and be manageable from the varies angles.

In addition to this, the small hospitals or specialist clinics could
lay the fire of competition at the knees of the larger hospitals. Such
a mechanism would be the same as with the finance industry, where there
is potential for (accredited) good causes to do investment in their own
way thus providing the state with a comparison they do not control. These
small time hospitals of a size small enough that bankruptcy does not cause
disruption, even if just a few though more is in theory better, could
provide invaluable ideas about what hospital services and specialists 
are worth in the free market. It also gives specialists and any health
care people a way out of the state controlled environment, although they'd
still need to get money for services from the health insurance system.

In theory they could also rent machinery and infrastructure from the 
state, in which case they'd be in (more of) the same situation as the 
other hospitals.

This appears to be more or less an exhaustive system that can work,
leaving no holes as far as I can see. A problem seems to be the pay
by the health insurance: if health insurance is only coming from one
source, the health insurance could dictate prices. The only way around
that for the medical specialists (and other health care workers, the
whole hospital/unit) would be to go on strike until demands are met.

Patients could steer these issues by going to more expensive hospitals,
if a hospital can set its own rates for services. If health-insurance
is differentiated by cost, for example as:
-> "One would pay 1% of service costs up to a maximum of $2000,-"
   => Then pay $100,- a month.
-> "One would pay .5% of service costs up to a maximum of $1000,-"
   => Then pay $130,- a month.
-> "One would pay 0% of service costs, never pay anything."
   => Then pay $150,- a month.
And so on, this way patients would have an incentive to go to the
cheaper hospitals without getting into too much risk, but if they
feel the cheaper hospitals are death traps then they will probably
choose the more extensive coverage and go to more expensive hospitals.
Thus stimulating a higher pay for the hospitals, who are then free
to decide where to place that money: with better equipment, or higher
salaries to attract better people (and so on.)

It remains to be seen if choosing hospitals can be done rationally.
I'd be the first to acknowledge that a hospital that has a higher
death rate and lower curing rate may in fact be the better hospital.
The problem being that for some reason it is always getting more
difficult patients, perhaps due to a worse local environment causing
people to be sick easier and heal slower. In that case the apparently
"bad" hospital might still be the best choice. Comparison is probably
a relatively complicated issue, but in theory it could all be researched,
and the people could understand it; the market system could then still
function. As always so it seems: everything is 10 times more complicated
when it comes to health care then anything else.