But we do!
"Turn it on it's head ; what if the USA had something like the BBC?"
It's called PBS. It gets about half its money from the US government, the other half from sponsors. Other than quick sponsor mentions before the start of a show, there are no commercials, and as for BBC-like programming-- it brought us a whole lot of *actual* BBC programming back in the day. Any BBC show which had a pre-mainstream-Internet fanbase in the US had it because of PBS.
Even with the Internet and BBC America in existence, PBS is still showing some BBC material-- Downton Abbey being the big name of the moment.