The dress reform/rational dress movement did tend to go hand-in-hand with women's suffrage; at the same time, because suffrage was such a large movement and dress reform wasn't, the majority of suffragists were not particularly interested in dress reform.

(I'm going to write mainly about the US and Britain here - I'm not sure I can really speak to the specifics of New Zealand in particular.)

Dress reform was a cause that was very dear to some hearts. It began in the 1850s, with Amelia Bloomer's promotion of what became called the "Bloomer costume" in her magazine, The Lily; this outfit certainly looks of its period, but it differed from fashionable dress in its lack of a corset underneath the bodice or petticoats/crinoline under the skirt (as well as a general lack of ornamentation), and the loose "Turkish trousers". It was absolutely loathed and ridiculed by the mainstream, which saw women wearing trousers as crossdressing and indicative of an intent to turn the patriarchy upside down - but it was also not popular among the nascent women's right's movement.

For one thing, the movement was mostly focused on women's lack of legal rights, such as their right to own property when married and to vote: the idea of challenging sexist customs as a major pillar of feminism is quite modern. For another, it was simply not aesthetically pleasing. Without a corset, the bodice had to be baggy and wrinkled in a way that you simply didn't tend to see in women's dress, even the dress of women who weren't hyperfashionable, and the skirt was likewise so slack and narrow. The trousers were loose in order to comply with standards of modesty, but that also caused the same problem. Most people care about their appearance to some degree, and it's a lot to expect people who otherwise agree with you to believe that their choosing to look "bad" will actually help your cause in the long term.

The "Bloomer outfit" died fairly quickly, as did dress reform more broadly in the public eye. It did continue, but while retreating into text and lectures rather than direct action. Women like Charlotte Perkins Gilman, who came to dress reform in the 1880s, could wear "corset waists" - bodices made like corsets, but stiffened only with extra strips of fabric - rather than corsets under their everyday clothes without actually looking like an obvious reformer. Still, even Gilman, who connected a stereotypically feminine appearance to stereotypically feminine failings, saw herself as needing to take a different path to do her intellectual work, and believed that women's clothing was dangerous to their health in a number of different ways, was not averse to dressing gorgeously and sexily when she felt like it. Even when she modified her clothing to be in line with what she considered healthy, she was still working quite hard to be attractive, and part of the reason she was successful was that she was thin enough to still look slender and taut without a corset. Ultimately, there was a contradiction in dress reform that few could really bridge.

This was recognized by leaders of the women's rights movement, which became largely focused on suffrage as the keystone right by the end of the nineteenth century. National and local groups might discuss the possibility of dress reform or ways to attract more women to it, but photographs of the women involved show that they all continued to present themselves in fashionable, corseted, long-skirted clothing. Mainstream dress was even weaponized, as marches for suffrage full of women in fashionable white gowns, shoes, and hats made it clear that this was an issue that needed to be taken seriously, rather than kooks who could be written off as fringe weirdoes (as they were usually perceived). It's difficult to do justice to this dichotomy in a museum context, and it's easy for curators to focus on the written statements about the horrors of fashion rather than balancing these texts against what actually tended to happen in real life (particularly as fashion history is very often served poorly in museums in general).

Ultimately, reformers fought against corsets because they believed that a) they were disastrous for women's health, in line with common medical opinion of the day, and b) they were analogous to women's role in society, an aggression heaped on women that was not required of men. There is still a lot of room for contextualization in the latter reason (the former is just wrong), but it was sexist for women to be required to wear corsets in order to be seen as respectable and worth paying attention to.