Not using a favicon API service like the once from DuckDuckGo or Google for these reasons:
APis are limited to doamin-level icon, whereas the web and browsers use per-page icons.
Windows really wants a 24*24 PNG icon which is the preferred icon in this implementation. APIs gives whatever size and format they feel like at that point in time. Web authors retain control over the icon used.
API usage leaks information to the service providers about the pages people visit.
The web is better when it's allowed to remain weird and decentralized.
Not using a favicon API service like the once from DuckDuckGo or Google for these reasons: