Although not strictly relevant to this episode, you might be surprised to know the first women to qualify as doctors, in many countries including the UK, US, and Canada, was actually in the 1800s. Not many of them and they had a hard fight, but they did manage to get educated, qualify, and practice. (And, of course, there is a very long history of women practising medicine in various capacities, but I understand your gripe to be specifically about the title).
To a certain extent, I agree with you. The lack of historic clothing does bother me, and I don't think we should gloss over the difficulties of women in the past, particularly given the rights women have today were hard-fought for.
However, I also think there's a real danger in painting the past with too broad a brush. The status of women and ideas about gender roles vary from era to era (there's a train of thought that the redefinition of gender roles in the Victorian era seriously harmed women, and they had more freedom and possibilities in the eras immediately preceding, for instance), and there have been always been outliers--women who through birth, circumstances, or inclination did unusual things. There were women who had power--in most cases soft power, but in some cases direct power. It's also worth remembering that our perception of women in the past is often shaped by upper-class women, because they tend to be the ones stories are told about, and the lives of working class women were quite different (the general rule of thumb is the lower you are on the social pyramid, the more freedom you have, although by the same token your life is more likely to be nasty, brutish, and short.) And although I'm bugged by the lack of historic clothing, the idea of what was 'proper' is hugely region/era/class specific (I admit to not being hugely familiar with the 17th century, so can't comment, but in the 18th, daring young noblewomen did sometimes wear breeches to ride, and the impression I've gotten is it was considered mildly racy, but not a social shunning type offence.) I think it's a disservice to ignore what women did and could accomplish.
That said, here's what we're told about Becka in this episode: a) she was born into what appears to be a peasant family, b) she married a wealthy landowner, c) said landowner is no longer in the picture so presumably she's a widow (I don't remember this being addressed in the episode, but please let me know if there was a line about it I missed.)
With this in mind, I have no problem with her cutting down a tree. It's certainly not something an upperclasswoman would be doing normally, and it's a bit eccentric, but as a peasant lass, likely she was accustomed to hard labour before her marriage, and there appears to be no one in her family who could stop her from doing so.
I also have no problem with her appointing herself chief inquisitor of the village because, well, taking care of the village would be her job. It would have been her husband's job, and in the absence of him having a son/brother/other male relative (which don't appear to exist), it would fall upon his widow. Remember that 'widow' was historically once the cushiest role for women, or at least women of rank and property. They no longer had to answer to husbands or fathers, they had control over their own property, in many eras standards of sexual morality were relaxed for them, and sometimes they inherited roles or responsibilities from their late husbands. (Note, as with all things, this does not uniformly apply across all eras and regions.) In this case, she exercised her responsibility for the village in the form of witch hunting.
And I have really abused brackets in the post.
