I apologize I was not more clear.
You gave two examples. One is using the favicon permission, which I believe you are referring to as the "favicon receiving API". This is the method that is used in the sample you linked to. This is useful if you want to know the
exact favicon used in a specific tab that is loaded. Many change their favicon based on the state of a person using their site. It may show information like the number of unread messages (as in the gmail example I gave originally), a different color theme, or any other kind of customization.
The other example (i.e.
https://www.google.com/s2/favicons?domain=google.com) which I believe you are calling "the old method", does not have the ability to access the state of a user's tab. It is just loading a url. So if there is any customization of the favicon, then you are not going to be able to get it using this method.
Does that make sense?