I am not excusing the BBC, but don’t think the comparisons with Trump stand up, because Trump is one man, and ‘the BBC’ is an organisation. A man can lie and be responsible for those lies, but an organisation is made up of lots of people, and is not an entity in itself. The BBC can’t lie - only people within it can do so.
I don’t think it’s realistic to expect the DG to be able to have an oversight of everything that is broadcast by an organisation as big as the BBC. I do, however, think that where it is clear that lies (well, mis or disinformation) have been told, the people responsible should be sacked. Maybe, if there isn’t one already, there could be a chain of responsibility so that programmes are routinely fact-checked and it is clear where the ‘buck stops’?
I don’t want to lose the BBC, as having a press that is privately owned and/or responsible to advertisers would be a retrograde step, IMO. On the other hand we have to be able to trust the integrity of a national broadcaster.