The chair has to win the support of the directors, and if the chair doesn't, the chair can be ousted from that position by a simple majority vote of the directors with little or no notice.
The chair's ouster is likely to happen if a chair doesn't have the support of a majority of the shareholders even if the directors initially support the chair, since the directors serve at the sufferance of the shareholders and could be removed if a majority of shareholders (especially in a closely held company) were dissatisfied with the directors' performance.
The chair is mostly a symbolic role as first among equals on the board of directors, unless a contrary contract is reached with this person. So, there isn't much power to assert vengefully. Even the chair's parliamentary procedure rulings could be overruled by an appropriate majority of the board of directors.
Normally, as a matter of custom and practice, a chair doesn't get appointed unless the nominee has the nearly unanimous support of the directors and shareholders, at the outset, at least.
In ordinary circumstances, almost all of a chair's power is "soft power" deriving from the fact that everyone on the board and in the senior management team respects the chair's judgment, not "hard power" arising from the chair's legal rights.
Frequently, the chair is a retired CEO, a sitting CEO, or is one of the largest shareholders of the company (or a representative of that shareholder). An outside chair who is a non-shareholder would be unusual and chosen only with the full backing of the directors and shareholders to clean house in a failing company in most cases.
A bad CEO or corporate president appointment is a much bigger deal, because usually a CEO will have an employment contract with golden parachute terms and will have much more formal authority to abuse.