You are absolutely right to be concerned about the importance of a good Chair/Manager relationship. However, this needs to be part of a system that has a clear and mutually agreed allocation of roles (who is responsible for what between the Manager and the Board as a whole). It also needs to be part of a culture of mutual trust and respect between the Manager and the Board (as a whole).
While it might seem a bit awkward at first, regular 'Board Only' time or 'In Committee' Sessions (both terms mean the same thing) are generally considered to be a part of good practice for a non-profit board. It's a chance for board members to meet and freely discuss without any staff or observers present. And if you put it in place when there isn't a major conflict, it's easier to use when things get tough.