More like "go woke because you've run out of ideas." The only "woke" thing about Whittaker was that the doctor was female, something they would have done in earlier episodes (in fact, did do) and wouldn't have made a big deal about it.
But they had no reason for people to watch Who by the time you got to Whittaker, and the show should have been cancelled already but it was making too much money, so they treated her like a Cousin Oliver and made the fact that she was the First Female Doctor (not even true) into a huge thing because they had nothing else.
I think the episode written by Malorie Blackman about some random time criminal appearing during MLK and Rosa Parks' protests would fall under "woke" too, but that was just a shit episode because of bad writing. A competent episode writer could probably have done something better with that time period/setting
Probably the fact that in one episode[0] the Doctor's incarnations were actually mostly women except for the most recent few - who obscured this important fact - smelled a lot like "yet another man taking credit for a woman's work".
[0] https://en.wikipedia.org/wiki/The_Timeless_Children