There are different questions poised in OP.
First, people have got to realise that they are wrong before there is even a chance of admitting they are wrong - many will never admit they are wrong eg as seen last week at Capital Hill certification interruption, Trump (maybe) but his supporters still believe the election was stolen so are unlikely to admit they/he are wrong.
Hancock perhaps erroneously?!., gave a false statement. Politicians lie,spin and weasel sometimes... but you are correct, that behaviour erodes trust further.