Have you run into this scenario before; a Board member resigns during their term and the Board of Directors (or Managers), collectively, appoints a replacement to the position that was just vacated? It should be smooth sailing and without issue, but if this is a cooperative or condominium with an opinionated ownership base, they’ll want to know why the new Board member wasn’t appointed by the shareholders or owners, as a whole.
While it would seem that putting it out to all shareholders or unit owners would be a great idea, it’s not often done, for a variety of reasons. One reason is that it is difficult to get all shareholders and owners to another meeting that is not the Annual Meeting and it is much quicker to appoint someone who is interested in serving on the Board for the remainder of this term. In some cases, there may be a committee member who has been serving on the periphery of the Board and this is a great opportunity to increase their level of participation, if it has been shown that they have something to offer, beyond just filling an open seat.
The reason that the Board will have the final say of who to appoint, in most cases, is provided for in the Bylaws of the building. For instance, below is a partial section of the Bylaws from a NYC cooperative, as it pertains to Vacancies:

When any vacancy exists or occurs among the directors by death, resignation or otherwise, the same shall be filled for the remainder of the term by a majority of votes cast at a special meeting of the remaining directors duly called for the purpose or at any regular meeting of the directors, even though a quorum shall not be present at such special or regular meeting.

As you can see in the above paragraph, the Board has complete control over who they appoint to fill the remainder of the term that was vacated. This could be a period of only a few months or it could be a multi-year term that the new appointment will fulfill.
Just because an appointment was made to the Board does not mean that the shareholders or unit owners are completely without hope if they absolutely disagree with the new appointment. If for any reason they are completely unhappy with this appointment, or the Board in general, the could also look to their bylaws to remove either one or more persons from the Board. They would have to follow the passage in the Bylaws that pertains to removal. From the same NYC cooperative bylaws on Removal:

Any director may be removed from office at any time with or without cause and at the pleasure of the shareholders, upon affirmative vote of the shareholders of record taken at a shareholders’ meeting duly called for that purpose; provided, however, that the directors elected by the holders of Unsold Shares can be removed without cause by such holders of Unsold Shares who alone will have the right to designate a replacement.

There’s no way that each and every shareholder or unit owner will be appeased in each situation, and this is a great example of this. More often than not, the Board will have someone in mind to fill a space on the Board if one should become vacant for any reason during the term of service. It’s up to the shareholders or unit owners to take matters into their own hands if they feel their interests are not being properly represented by the existing Board members.
As I always say, if you’re unhappy with the Board you have, throw your name into the ring the next time around and get on the Board to make a change.