They are trying - but the funding is more or less fixed (bar annual inflation) until 2022, and the charter is in place until 2027. I suppose if there were sufficient safeguards against government interference the money could be provided from general tax, but there would still be complaints that people who didn't watch/listen to the BBC were paying for it - let alone people that didn't watch any programmes on any device.
And there are complaints from people who drive that their road fund license doesn't actually go to the roads, and complaints from people who home school or privately educate or don't have kids that they don't get a rebate for the education they aren't using - that's the nature of central taxation, and you vote for the candidates and parties that best reflect your preferred spending options.
Blue Planet, Blue Planet II were exceptions, getting high viewing figures. In general documentaries are beaten by entertainment or "reality" shows.
That they are popular doesn't make them any different in intent from the other documentaries, and the documentaries (and, more importantly, the news) aren't intended just for the elites. It may be that people who don't think of themselves as 'elite' aren't inclined to watch them, but that's self-exclusion, not intent on the part of the BBC - maybe if more of them did watch we'd have a more enlightened society?
I think so, which is why I'm in favor of the current system with a review to see if it can be brought up to date with consideration of the purpose of the BBC and newer technologies.
My absolute preferred option is keep the BBC doing what it's doing, allow it the freedom to properly explore streaming (which it's tried to do, but been prevented from by the terms of its current charter, as I understand it), and just change the funding, but I'd be willing to accept it having the commercially viable portions of its programming stripped away to preserve and protect the factual programming element.
Having said that, I'm mindful that 'Top Gear' was (maybe still is?) somehow classified in 'factual programming' so the definitions might need tightening a little as well...
O.