What a great question! Let me do my best to answer.
In the United States, due to the electoral college, you actually vote for the electoral candidate that will vote for the person that you want to be president. So, that is perhaps the closest to your question. (However, you can get into the "faithless electors" problem and that's a whole different kettle of fish, so to speak.)
In most parliamentary systems, you would vote for the person or party that you prefer. It does, as you pointed out, limit the ability for you to vote for the people from other parties that you think would make excellent choices. (For example, if you overall like the Labour Party in Great Britain, but think that the Green Party has a candidate that would be an excellent Minister of Transportation, it does make it difficult.) That's where coalition governments are quite helpful; oftentimes, most parliamentary systems do not achieve a majority of the vote, and so they make alliances with the other parties, so you may get to see your person in action!
In many African countries, there is a difficulty with getting leaders to relinquish power in the post-colonial era. We've seen this in Zimbabwe and Mozambique, and that's a much more serious problem, where coups and violence can become more realistic scenarios. There have also been these problems in some of the socialist countries of South America as well.
Far Eastern/Asian states have complex systems of election, some still maintaining monarchies alongside democratic republics (Japan, Indonesia) and some that are completely closed to free elections (North Korea). Middle Eastern states are largely monarchies.
So in answer to your question, it is rare to have the group elected and not the person to a state of leadership, unless it is in a democratic republic (such as when the House or Senate turns over to one party or the other in the United States) or to a large majority or minority in the parliamentary systems in Europe. I hope that this answers your question!