[*] All parts of the Israeli population have high birth rates, even the secular Jews. I find it misleading to single out Orthodox Jews as the main contributor. You don't do that to Evangelical Christians or Mormons, or some other groups in US.
University students are predominantly female in Israel as in North America, so female education isn't a differentiator for fertility rate.
I recall reading an article about the culture of Israel is friendlier towards families/young children. Not sure how that cultural aspect would show up in stats.
- As of 2022, the fertility rates in Israeli cities dominated by specific demographic groups were: Haredi 6.1, Bedouin 4.4, Jewish non-Haredi 2.4, Arab 2.2, Druze 1.8
So the GP's characterization is correct.
Yes, non-Haredi Jewish Israelis have a higher fertility rate than in Western nations. There seems to be a variety of reasons for this including cultural/religious pressure, institutional support and selection bias (in terms of who chooses to immigrate/emigrate).
The OPs (incorrect in this very minor detail) point was that fertility is below replacement everywhere except Israel, where the only reason it wasn't below replacement was Orthodox religious members of society. That's just wrong. Fertility in Israel, across every single group except for one in your own data, is above replacement. Orthodox are the highest, and bringing the overall number up, but if you removed them, fertility in Israel would still be comfortably above replacement.
So no, their characterization was not correct, since the point was that, absent a particular group, fertility would be, like in almost every other OECD (and many non OECD) countries, below replacement.
I don't think the religion is the driver here.