Honestly, what that study found seems kind of intuitive to me. I think those surprised about religious men doing more are stereotyping current religious men/families with the more rigid patriarchal model that was popular many decades ago both with religious couples and culturally overall. Over the past 30-40 years, the family model that has been growing the most in the church tends to maintain some traditional family leadership structure but also has taught about husbands serving their wives, which includes helping with what they would see as her traditional roles. Look at any of the main sites for that grouping (Focus on the Family, xomarriage, etc.) and you'll see a lot of advice like that. Some odd concepts have come out of that, like "choreplay", but that's another topic.
The egalitarian, non-religious couples clearly makes sense as well as their entire worldview centers around an equal leadership and role structure for the family.
So that leaves the groups between those two, who don't have something in their worldview (like faith teaching or equality focus) to drive changes - and that means they more naturally go with what they saw their parents and grandparents doing. Result is they would lag in changes to who could do what chores, etc.
