As I've said in chat, I believe questions can use up to five tags, and there is nothing wrong with using all five if there are five tags that are appropriate for the subject.
In this case, both chile and santiago-de-chile would be fine, and adding them during the initial cleanup of a new question is appropriate. If the question was years old and not in any particular need of editing, I would not retag it.
Attempting to distill everything down to single hyper-specific tags is unhelpful, as this practice is not intuitive to askers and new users, does not help the "related" feature to show the most relevant related questions, fails to surface questions to topic experts who may follow particular tags, and makes tag badges and the golden hammer that comes with it impossible to obtain or use.
That last bit about the golden hammer is important. Users have gone to considerable trouble to earn, say, the uk gold badge and can use their power to rapidly clean up (or reopen) duplicate questions. This lets, for example, Gayot quickly address many UK visa questions, many of which can be addressed by duping them to our canonical answers. If you insist on removing uk tag from such questions in favor of only using standard-visitor-visa or other more specific tags, this site feature becomes useless.
tl;dr: Use country tags when a question reasonably relates to one or more specific countries. Then add more specific tags as needed. Do not use country tags when a question is suitably generic that the country is irrelevant.