> an extremely arrogant demeanour instead of the modest gait and bearing proper to the gentle sex
That is, the 'Gesta Stephani' is claiming that she should not be arrogant because it is not proper, because she is a woman.
That evidence?
We might reasonably infer that she got called arrogant for doing something that wouldn't be considered arrogance in a man, the article has unhelpfully elided that detail (maybe the original author did as well). But we've also got to admit she might have actually been unusually arrogant, that does happen from time to time. If I were in charge of a military I'd be standing around even more arrogant than I already am.
pjc50 offered a good example of something that was actually gendered - women not being able to own property in their own right for whatever unfair reason. The quote isn't that sort of out-and-obvious sexism. Said quote could almost be modern political whinging.
PS I'll add in an acknowledgement that words did mean different things back then so possibly at the time the only reasonable reading was with an "[as with all women]" on the end, but the fact is that as written this quote is making specific accusations of arrogance and arbitraryness that are not contained to either gender and are bad in all leaders, female or otherwise. He didn't write "women are all arrogant" or "women are all arbitrary". If anything the thrust of the quote is most women weren't either of those things. It doesn't support the idea of double standards or ludicrous criticism that would never be made of a man.
To address your question, though, yes I absolutely read "proper to the gentler sex" to be a statement about how Queen Mathilda _ought_ to have behaved. Also I have strong doubts that men (especially male rulers) at the time would have had the same standard set for arrogance. Of course a King could be in fact arrogant and then critics might point that out, but I do not see why they would bring his sex into it.
One thing we might agree on is that all political criticism can be false, and whether or not she was _actually_ arrogant is difficult to say. In the context of a bitter civil war, any record of events will have the strong bias of the author. The modern world did not invent fake news.