The example you cite is not only ethical but normal in American non-profits.
 It's how many of us keep good things going!  Many (possibly most) 501(c)3
boards are "self perpetuating" via governance nominating committees not only
in museums but in other areas. First-hand examples I'm familiar with, include
social service agencies, universities, neighborhood associations, alumni
organizations, travel and tourism councils, historic preservation agencies
&c.  Often, other non-profits actually officially sponsor the creation of new
non-profits which, in turn, become self-sustaining through common communities
of interest.  With roots in de Tocqueville's observations, we've long used
this mechanism as the ways and means to get good  things done in a democratic
society.  Pretty empowering stuff!

Disclaimer:  as empowering as this system can be, politics and personal
agendas often upset the opportunity afforded by this sytem!  In any case,
good luck to you.