Many people do not even bother to look at indexes. However, in academic circles, indexes are extremely important.
You hit the nail right on the head when you bring up the topic of specificity. If you think about it, if the index is too specific, then the index would be enormous and might not be as helpful. And if the index is not specific enough, it would provide little help in terms of looking topics up. So, the goal is to have an index that has the right amount of specificity.
The general rule of thumb is as follows. Indexes should cover all main topics and themes. So, if there is a book about the Roman Republic, an good index should have topics on all the major figures and places, broader topics such as religion, sacrifice, economy and the like. Minor figures who are barely mentioned need not be included as well as minor details.