You tell that each clinic is managed by 2 heads: chief doctor and manager. I do not believe in shared responsibilities. One has to be on top of the other. As your 5 top managers are not physicians and you have a health-care organisation, I would have chief doctors managing the clinics with the help of financial and other staff. As your dentists are the key persons delivering the high quality services, I would listen very carefully the ideas of chief doctors and other dentists how to develop your services and business. And then reciprocally, the chief and other dentists are to commit to your financial goals.
Private equity owners do not always understand what is doable at an expert organisation with high quality services. If I were you, I would carefully review their goals and estimate which goals are realistic to reach. For the other ones I would have a bold discussion on whether there is a need and possibility to change them. Once your own goals are in line with the board expectations, I am sure you will manage to lead your organisation towards your own vision. But first you have share the values and goals of the board or convince them to adjust the goals. After that I think merging 3 cultures is still a minor task compared to the original board expectations.
I think the Module 3 paper “Transforming Care at UnityPoint Helath – Fort Dodge” by Amy Edmondson et al nicely describes the same problem. To create patient-centric care you have to break down the silos and design coordinated care. To start with you need a steering committee consisting of representatives from all stakeholder organisations (ie hospital, home health etc.) to lead the change. The same stakeholder organisations build working groups to redesign the treatment/care processes. Co-ordinating teams and a patient navigator are good examples how to to smooth the patient path. Also at our organisation we have good experiences to use Lean tools like A3 to improve our services and cut the waste. Having a special palliative care program helps to serve well also the sickest and support their families.
I think it usually is more difficult to train a director with excellent communication skills to become a highly appreciated technical or other expert in the field than to give coaching on communication skills to an outstanding expert. I think you should keep the technical director but her boss should latest now have a constructive discussion with her on these problems. The boss could in the beginning praise her with her excellent technical skills but be frank with the problems which may endager her success in her career also later at any other company. A mentor or coach is an excellent idea. Maybe her bonus could be bound also to her improvement in communication skills.