To the best of my understanding, DW is actually doing pretty well in the current TV climate, still ranking as one of the highest watched shows. The crucial context here is that all TV ratings are dropping, as these older platforms simply can't compete with the new, digital, streaming, on-demand ones that can throw tens or even hundreds of millions of dollars at their shows.
As to 'woke', or 'sjw' stuff, it's near-impossible for me not to roll my eyes when people start throwing those misappropriated terms around. The Doctor, by the character's very nature, is a Social Justice Warrior; the phrase defines what they do, always have done, and always will. The current incarnation is no more or less 'woke' than the prior two, even with a female lead; the difference is how poorly a lot of it has been executed, from slightly clumsy, like Rosa, to downright painful, such as Orphan 55. The writing is the issue, not the themes, and certainly not a frustratingly under-served Jodie Whittaker.
And, with the return of Russell T Davies, whose first run introduced Captain Jack 'I'll flirt with anyone' Harkness, and whose other credits include Queer as Folk and It's A Sin, 'woke' Who's not going anywhere any time soon. If anything, it'll only get more so, just (hopefully) executed a whole lot better.