They need to make that more well known then. As a consumer who reads the papers, ....ere the money goes and what it's actually used for I might be willing to pay.
They do actually try to make it known to the general public, unfortunately they're not allowed to take out full page adverts in the DM to say "this is what we do", and the papers love distorting what the BBC does do.
For example they'll go on about the huge salaries of BBC employees, usually forgetting to mention that the BBC is paying less than ITV or C4, let alone Sky, and frequently taking the amount paid to get a ready to air program as the personal pay.
Or they'll talk about the BBC sending people on "jolies" forgetting to mention that they were working, as an example of that one of the Doctor Who Confidentials had Russell T Davis and the then Doctor in Venice, the Mail ran a story about the BBC spending hundreds of pounds to send them there for fun, what actually happened was that as they had a film crew in Venice taking required shots for the actual Doctor Who episode, they decided to get the full monies worth out of the film crew by also filming background stuff that could be shown on the secondary show, so for the cost of a couple of plane tickets they got an extra 15-30 minutes worth of content (which would have been a bargain given the cost of the film crew).
What the BBC does do is put up things like blogs and pages on their website explaining some of the technical work they're doing, or have implemented, from memory one example of that was recently they did feature on how they did testing for ultra hi def broadcasts using a prototype encoder built in house using commercially available multi function chips as proof of concept, or the work they did to replace aging broadcast equipment for Radio 4 (I think it was) that had to maintain compatibility with data broadcasts as there wasn't a commercially available replacement.
IIRC they're also for example doing a lot of work on video compression (something they've been working on since about the 70's when I think they developed an early digital video recorder), with things like improving h265/HEVC for broadcast and production usage.
I think they also offered Iplayer's tech to the other broadcasters at one point, but were told they weren't allowed to because it would have hurt the commercial companies making their own versions (so ITV player could potentially have been much better
).
Their accounts are also readily available if you're interested although you have to remember little things like if the BBC pays for an hour of content, it will be about 58 minutes of content whilst ITV might pay for an hour of content and get 45 minutes worth (due to adverts).
Which means that on paper the BBC content may be more per hour (because it's got 20% more per hour), but in terms of cost per minute of produced programming it's cheaper.
Likewise BBC radio is more expensive than most of the commercial broadcasters because the BBC actually employs people to be on the station, and for content on the station (the reason R4 is so expensive is because it has actually programming rather than just a DJ/announcer).
A lot of the commercial broadcasters might only have one or two people running half a dozen stations much of the time, and only have live presentation (recorded for all stations in the network) for a few hours a day.