Why do people need things like Google Analytics when they can just self-host? I thought the reason people rather use actual Google Analytics is Google's ability to track specific people across multiple websites and promote websites with Analytics in the search results. But 3-rd party alternatives can't do this anyway or can they?
Yes. There are a number of powerful self-hostable open-source website analytics solutions, for example https://github.com/matomo-org/matomo.