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The answer is: Probably not.
I would err on the side of caution and say no, an executive director should not be an independent contractor. This is because they would likely fail multiple questions of the IRS’s 20 Factor Test. The most obvious one that comes to mind is #3 on the 20 Factor Test- Integration. The IRS says that if the services that someone is providing for an organization are vital and ongoing for the organization’s success (i.e. they are providing continuing day-to-day work) then they are probably not an independent contractor.
The only circumstance that they might be able to be an independent contractor is when they are serving as an acting or interim executive director.
To understand more about the difference between independent contractors and employees, read Independent Contractor vs. Employee Part I.
To learn about the process of transitioning an independent contractor to an employee, read Independent Contractor vs. Employee Part II.