It's rather unfashionable at the moment to say you're English or fly the English flag, mainly due to people like the English Defence League.
Nah, the EDL haven't been around long enough. It's still unfashionable but not as bad as it was a decade or so back when just showing a St Georges flag suggested you're a white supremist (the logic seemingly being that the flag is mostly white) and that it offended minorities (muslims in particular) although the media seemed to have difficulty getting anyone from a minority background who held this opinion (very much in line with the 'Christmas being offensive to non-Christian minorities' assumption). It's a bit easier these days; you're not necessarily a white extremist if you have a St Georges flag outside your house but merely a low class bigot, like the poor white van man in Rochester whose house is now a national landmark.
England's also a constitutional inconvenience. It's just mainland or 'rump' Britain to the establishment and not something distinct and separate like Scotland or Wales. Likewise, whilst 623058 may live in Scotland, me and you live in North West. That's how Whitehall draws the map and how it runs the country.
Thirdly, there's no BBC One England. The Network feed has to have presentation that may be potentially seen across the whole of the UK, even if only 'rump' Britain (i.e. the parts that are not Scotland, Wales & Northern Ireland) sees it most of the time. Also, the Channel Islands and Isle of Man are not England either.