I think the whole "Kirk is a renegade rule-breaker" thing only really started with TNG, where in comparison to Picard he can seem like that (even though he isn't). Both Kirk and Picard respected the rules and laws, but they go about that respect in different ways. Picard will sit in his ready room, calmly debating a law/mission/rule with one of the cast. Kirk reaction is generally a bit more passionate, but he never oversteps his bounds.
I think that difference is overstated. Remember "The Drumhead," when Admiral Satie called attention to how many times Picard had violated the Prime Directive? The two captains may have had different styles, but they weren't that different in substance. Sure, "Unification" had that line about Kirk's "cowboy diplomacy," but that was in the fifth season. By that point, the movie version of Kirk as a renegade was pretty well established.
And now that I think about it, Picard became something of a renegade in the TNG movies too. In
First Contact he defied Starfleet's orders to stay out of the battle, and in
Insurrection -- well, the title speaks for itself. Maybe it's just that heroes defying their authority figures are such a common trope in movies, e.g. all the Tough Cops Who Don't Play By the Rules. I wonder if there's something about that format and length of story that lends itself to that trope.
A misconception I had about the show was that it was progressive. And to be fair, it was in terms of racial politics. But when it came to women... it really didn't do a good job there. In fact it did a blindingly terrible job with women.
In retrospect, certainly. But for its time, it was relatively progressive, though backward compared to several of its contemporaries. TOS's fan base consisted predominantly of women, and the first generation of fanfic writers were overwhelmingly female. True, to some extent what motivated them to write fanfic was the desire to correct ST's shortcomings in its portrayal of female characters. The reason the Mary Sue trope came into being was because so many authors felt compelled to add new, capable, important female characters to make up for the lack of same in the show's ensemble, and naturally some of them took it too far or did so hamfistedly. But still, there was something that drew them to the show in the first place, that made them feel it offered inspiration for female fans even if it fell short of its potential.