You've mentioned home education groups, it may be worth discussing with the parents in them how they balance their home life. It was a popular topic in some of the groups I was in.
IME, it varies a lot and there is no norm. When I was home educating four kids, I knew families where mothers did everything as you describe while the father works, I knew single parents who home educated doing everything with some support from friends or extended family, and I knew couples and separated co-parents working opposite shifts to make it work. My husband and I did various ways of working to balance it all over the years.
Well before we had kids, my husband was doing a significant amount of the housework including most of the cooking. From our first, my husband would regularly take them for walks and trips out while I had some quiet time and was settling the kids to bed from around 6 months. There are times we've needed to rebalance, but even during times when only one of us has been working and/or one of us was doing nearly all the housework, we've had discussions around being tired, drained, but not throwing a lack of gratitude at the other party for discussing difficulties. Life is full of difficulties, part of being partners is helping each other work through them, not to belittle each other for struggling.
The various ways home educators make this balance isn't going to help if you're in a partnership with the kind of conflict that's being described. You'll need to hash out what kind of balance you each want with each other.
There are a couple things that are confusing:
We both have thought I could provide a better education for them than at public school. It was my initial idea
Why public schools specifically?
I would rather home educate my children than go out to work yes. Is that a bad thing these days?
'These days' - home education hasn't been the norm in most of the UK in well over a century, for some groups far longer. Even with the post-COVID jump up in numbers (which still baffles me with how badly home educated kids got screwed during that time) it's still a small minority.
A lot of people prefer being with their kids over working, that's not the same as comparing working with home educating. I quite enjoyed home educating most of the time, but it's full on and I don't think most people would want to be fully responsible for all aspects of their children's education, social needs, and more if they have other suitable options especially with older, secondary age children. That's why a lot of home educators are those with children who have additional needs who either have issues getting a suitable placement in the first place and/or have had severe issues in the school system and were withdrawn.
Why would a father not want to get up in the morning with the children he wanted?
You should ask your husband that.