Her very first words to me was " your breaking the law not wearing your highviz !"
We had a similar situation at my work, a new 'boss' came in and was going round stepping on everyones toes - I don't mean metaphorically - actually stepping on peoples toes to check they were wearing steel-capped safety boots!
I do sometimes shake my head and say to myself "its H & S gone mad...." however I wouldn't like to work in other parts of the world (or Britain a hundred years ago) where workplace injuries and fatalities are frequent. Having said that, some 'managers' just love to use H & S as a tool to weald power.
In respect to the op and this persons particular problems: I would be very cautious of bringing the gender issue up when any grievance is made. Stick to the facts and the mistakes made and also make sure you yourself stick to company policy (eg the highviz rule). How can you be critical of others if you are not perfect yourself? It might very well be a 'box-ticking' recruitment decision possibly because in the past golf-clubs had something of an issue with being male-dominated environments* But don't bring that up, you can't in 2022.
*I was a member of one in the 1980s, there were two bars one of which women were not even allowed to go into - except staff of course, to serve tea and sandwiches!