I'm going to play nihilistic devil's advocate here, because it's actually my position when I really think about it*.
I dont think we are better off with the factual inaccuracies, because depending on what they are, they can cause people to draw conclusions about current events etc based on false information, as happened with the JFK movie for example.
I'd argue this is more the exception than the rule, and I'd argue many of these people that jumped to the conspiracy would have likely done so with or without the Stone movie, at least the ones that were vocal enough to have an impact on our day to day lives.
Theres also the characterisation of historical characters which comes with inevitable and really terrible editorial bias, because they have to make a decision about the few things this character will represent.
This can then serve to make us more or less sympathetic to a position and detracts from the facts speaking for themselves. Sure, it can be used for things we deem positive, but it does so at the expense of the reputations or memory of real people and can equally and just as often be used for things we may not agree with.
So what? What does my exemplar Joe Blow the bank teller knowing about a specific historical event matter? If someone is genuinely interested in history and in a position to do something beneficial to the world they'll end up studying it in school, getting as accurate a representation as is possible, and using that information to further their work in academia. Joe will rant to his buddies who will probably not really care one way or the other.
I guess its good that people like Hawking and Turing and Dorothy Vaugn (who I had to google, to my shame) get the recognition they deserve.
So what? The people you mention are dead. I guess it's nice for their families that they get recognized but what's the ultimate benefit beyond that?
Historical stuff is (for 99.99% of us) of no more use than any other form of pop culture that isn't based on fact other than that it makes us feel smart and cultured in conversation. That's my argument. I would argue there is an insignificant number of individuals who were inspired by historical tv/movies into their career paths and that any inaccuracies found in the inspirational media are ultimately irrelevant.
*If I wasn't thinking about it deeply I'd rather everyone knew about important historical events and knew them as accurately as possible.