Numbers 30 deals with Israelite vows to God, i.e. declaring something to be dedicated to God. This would usually involve vowing something to God that one would give to the tabernacle or temple system much later, e.g. a certain percentage of the harvest that hasn't arrive yet or something of that nature. Some of the Pharisees in Jesus' time abused this system by vowing things to God that were necessary for caring for their parents, and thus they used these laws to get out of more important ones like honoring father and mother.
Jephthah in Judges 11 vowed whatever first came through his door, and it turned out to be his daughter. In that case, he tragically honored his vow when he shouldn't have done so, although if he had broken it he would have needed to make atonement. But other vows could be rash and should never have been made that nonetheless have to be honored. Typically if a man made a vow, he would have to honor it even if it was rashly made and burdensome to honor.
The regulations in Numbers 30 relate to girls and women making vows when under the authority of someone else. Normally a man would be responsible for his own vows. A girl under her father's authority would also normally be responsible for her own vows, provided that her father, when hearing about it, said nothing. But he did have the authority to cancel her vow. The same is true of a husband of a married woman. The father or husband would have to cancel the vow immediately when hearing about it, but the authority to cancel the vow came with being the father of a minor girl or the husband of a wife.
What interested me in reading this chapter recently was how it treats widows and divorced women. There were cases of widows and divorced women going back to live with their father, but there were also cases of widows retaining the property their husbands had inherited and serving as a head of household. These cases would have to have involved children, since otherwise the property might leave the clan, and property ties to tribes and clans was a very big deal in ancient Israel. But what's notably absent in this chapter is any statement about such women being under the authority of their father in terms of vows. As far as this chapter is concerned, a widow or divorced woman was simply responsible for her vows, and no one had the authority to cancel them.
What this means is that there's no underlying assumption that women can't make responsible choices and need some man to oversee their vows. Widows and divorcees would have to fall under someone's authority if that were so. Here they don't. There must be some other explanation. Complementarians have one. Women and men are not in their natures different (or at least not in their natures different enough) to justify women always and in every case being under authority and in need of supervision for their vows. But in the case of girls under their fathers' authority and women under their husbands' authority there's a role distinction that has nothing to do with men's and women's natures.
Some complementarians might extend something from this particular role distinction into our current day, while others might take it to be tied to the old covenant sacrificial system and thus no longer operational in any sense. But either would be able to say that it's based in a more permanent role relation that does continue today. Egalitarians, on the other hand, might have a harder time dealing with this passage. I'm open to suggestions, but I can't think of an easy way to take this passage given egalitarian principles. There might be some complex hermeneutical explanation for why it's not a moral mistake on God's part to allow women to be subject to men in this way, but the complementarian explanation is much more straightfoward.
It seems, then, that this passage fits best not with traditional patriarchy nor with full-blown egalitarianism but rather with the complementarian view that men and women can have different roles (in terms of authority) that nonetheless don't reflect any difference in nature. As I said, I'm open to suggestions about how egalitarianism can handle this passage, but short of that I'd have to say that complementarianism offers a better explanation of this passage.