I think you illustrate the Beebs main problem perfectly, by making the same mistake they seem to make. Firstly, you say the Beeb should " fill a void, providing news and programmes that the market won't make". And I agree, that is what it is there for. But then you say you "want it to...Broadcast, rather than narrowcast to niches".
Surely this is something of a dichotomy. If something is of general, broad interest to the population then it is something the commercial channels will be making or, if they haven't thought of it yet, will want to make. After all, they want to sell advertising to as many people as possible. The voids, the gaps, the spaces in which you seem to believe the Beeb should be working are, by their very nature, niches.
The Beeb does the same thing. It seems to judge it's own success by viewing figures and "reach" when you could almost make the opposite argument. Once a program or format becomes successful, the commercial operators will always copy it. Arguably, at this point, the Beeb has carried out its function and should stop and move onto something else.
Why does the BBC, an organisation whose main function is to make programs nobody else makes, make Eastenders, a southern copy of Coronation St? Why is it allowed to bid for football rights when that uses public money to inflate the value of those rights which are then sold to the public? Why does it have two mainstream music radio stations?
The BBC needs to be cut back hugely and should not be allowed to enter segments already catered for and should be forced to leave segments if it succeeds in creating new ones that come to be adequately served by the commercial operators.