The first thing to understand is that Israel's parliamentary system of government is quite different from the US'. Instead of two main parties dominating the political landscape, i.e. Republicans vs Democrats, Israel has around a dozen main parties which work together to form coalition governments where more than one party is in power at one time. Under this fragmented landscape, if you have a religious or ethnic minority who all tend to vote for the same party, that can lead to a situation where one group in society may hold sway over others politically even though they are not as as large a population numerically.
And yes, you're quite right - a lot has changed since 2016, and some Haredi (especially the young) have been increasingly drifting towards the far-right nationalist parties like Likud (Netanyahu's party) and the Religious Nationalists. However, as this AP article suggests, these recent converts moving away from the traditional Haredi parties are still a minority.
While the majority of Haredim living outside Israel still do not identify as Zionists (as per this recent, post-Oct 7th survey), I admit I don't have any hard polling data for the current situation in Israel itself. If anyone else does, I would appreciate the info.