Released today, Faviconographer is a “hack,” of course, but one that does a great job of serving the single purpose of adding those icons to your tabs. It’s not as clean as Chrome’s built in solution, but if the lack of favicons is one of few things keeping you from using Safari, it’s a good solution.
The app runs in the background and uses the macOS Accessibility API. It sends no data about your browsing (it doesn’t even save it to disk). It just waits for tabs to load, grabs the favicon, and applies it to the tab in the tab bar. There are a few limitations, but it works.
Faviconographer is free. You can read a bit more about the backstory and motivation, as well as grab the download at faviconographer.com.