See #3 and https://core.trac.wordpress.org/ticket/32340
The site name takes you to the front-end of a website, not wp-admin. Per @stephdau:
That site name was originally thought out to become a menu once clicked, giving site switching options to the user. This wasn't implemented (yet?) but the link stayed. I think going to wp-admin is a better link as well.
Q: should we make it filterable, while we're at it?
See #3 and https://core.trac.wordpress.org/ticket/32340
The site name takes you to the front-end of a website, not wp-admin. Per @stephdau: