In terms of OPs original post - I do think psychiatry is misogynistic - because society is, and historically has been, so psychiatry will naturally follow. This doesn't necessarily mean that men will be treated better by psychiatry, but that treatment will be influenced by sexism. Also psychiatry can be/has been used as a tool of control, so it will have (or have had) elements that discriminate against the group without power (women, black people, gay people, etc).
But in general ... there are FAR bigger issues than sexism! The whole understanding of and framing of MH issues, behaviour, trauma responses ... psychology seems to have an idea but psychiatry and thus MH services have lagged far behind.
I was pondering recently (I still sometimes feel overwhelmed, angry, and sad at how I was treated and the knock on effect on my life to this day) one particular aspect of all this. Related to the idea of what a "normal" response is, and the idea of taking into account the person's social circumstances.
I had the misfortune to have been (mis)diagnosed as having BPD/EUPD. Therefore every issue I had involving other people was seen as my fault, resulting from me being difficult and a terrible person. They completely failed to take into account that if you are traumatised and vulnerable, abusers are drawn to you. Plus you don't have the same resources around you (eg. financial, or solid trustworthy friendships) so you end up in more dodgy situations (eg. homeless or housed in upsetting/difficult situations) or having to rely on people you'd rather not. As well as perhaps needing to be encouraged to believe you deserve to be treated well and taught to spot red flags! (Rather than be told it's all your fault which makes you bend over backwards to accommodate abusers and chancers.)
Just one example of psychiatry majorly fucking up, where sexism is only a side order.