I think your view of the purpose of a noncompete agreement may be a little too narrow.  While I agree that charities should not be motivated by profit as such, they have to run their operations so that they can be sustainable and that requires ultimately having more income than expense. 

I previously mentioned hospitals that have had noncompete agreements with physicians for whom they substantially subsidize the physician’s training and the development of a practice in the community.   I have seen noncompetes used by much smaller charities that train counselors where the charity doesn’t want them to take their newly acquired knowledge down the street and compete with the original trainer.  Neither of these organizations wants to see their substantial investment lost just at the time it could be most productive for the organization and beneficial for the community.

(I know “compete” is a bad word to use with charities, but to say it doesn’t exist is to deny reality. And in a capitalistic society that loves competition, it may not be such a bad idea to have charities compete among themselves to provide better and more innovative services.)

A noncompete has to be based on adequate consideration on both sides and must be reasonable as to time, scope and place of its coverage.   But I think it is inadequate to look at a noncompete only as a way to protect a “profit” for the charity.  I think it is much more appropriate to look at it as a way to protect the sustainability of the organization and its ability to continue to provide charitable services to the community.  Looked at from that point of view, there can be many other situations in which you can imagine that a noncompete commitment can be reasonable for both sides.

Update:   Following publication of this question and answer, I was advised that a bill, H.B. 514, has been introduced in the Pennsylvania Legislature that would deny "purely public charity" status to any charity that enters into or tries to enforce a restrictive non-compete agreement with any employee. The effect, if passed, would be to deny state real estate or sales tax exemption to any charity that did so. The bill would not affect federal tax status, but real estate and sales tax exemptions can be significant for Pennsylvania charities.  —Don Kramer