The BBC gets complaints from both the right and the left because they have their own political agenda. They dumped on Corbin because he was for Brexit and they opposed it. The BBC is full on identitarian left wing politics, anyone that does not support that will get the BBC on their case. You might think the TV programming is safe but it is seen as a joke in most cases that is just pushing politics and people are just turning it off.
The BBC does produce some good programs but that is fading away being replaced by agenda driven garbage.
There are ways to save the BBC, and this is what I would do.
First would I have a proper comaplints process, as it stands when you complain about the BBC you complain to the BBC. There needs to be a third pary politically neutral organisation that deals with the complaints and it should have the power to impose really large fines on the BBC for being politically active.
I also believe that all articles on the BBC site should have a comments section and an ability to thumbs up and thumbs down earch article, like you get on youtube. This will allow people to show support or not about the articles and allows readers to see if others have problems with what they are getting.
Enforce a diverse set of opinions by forcing the BBC News channels to show news reviews programs from third party organisations.
Have the political makeup of the BBC public knowledge. There are political tests out there like the political compass
https://www.politicalcompass.org/test and 8 values tests
https://www.idrlabs.com/8-values-political/test.php every one publishing news on the BBC should have this information about them shown. It doesn't need to be these political tests but there should be something that allows the public to know the biases of the reporters.
Outlaw completely their campaining, it doesn't matter what the issue is they should only be allowed to inform people and if people then decide to campain that is it.
There is a major trust issue with the BBC and if they don't fix that then they deserve to fade.