The issue is certainly recognizable. The difference of 10x versus 4x for independent or employed physicians seems rather large but most likely you would otherwise not be able to compete with the top ten high salaries independent physicians. From the reimbursement side one could use cost pricing method to reimburse but this would allow efficient doctors to earn more but probably not ten times as much. For the employed doctors one could introduce a hybrid system recognizing that high paid doctors cannot do without the less paid doctors at some point and financially recognize the low paid doctors for that.
Why would a physician wants to perform dull work unless he or she is payed very well for production rather than performing on the top of their skills. I like the idea of value stream mapping to make a change.
We have the same issue in The Netherlands where data sharing is even mandatory between proton institutes. Legally I think in your case the Hospital owns the data. This is the same. We have chosen to leave the data at the hospitals and configure a center steered by the scientific community and a scientific board that can retrieve automatically answers to questions/queries on the data. In other words: the data is not leaving the hospital but the query results are. This works for discrete data but also for more fuzzy data like changes in grey-value for a given circumstance.
Almost working in Belgium one of the fields where one might win is content. Why are people leaving (doctors) or is there a shortage (nurses) could be turned around in why are people staying. This might be on the content side rather than on the financials. One way to get out of the trap of doing everything and for a fixed price is to encourage people to increase the fun in their work. Give them opportunity to be the best in their trade within the network by specialization of the various hospitals with cool possibilities for the staff you want to work for you. This gives financial room as you do not need to cover all in every hospital. Remember doctors and hospital: they are all in the same trap.