Yes, I'm afraid I agree with this too OP.
Some of the illnesses you describe should absolutely necessitate your DH stepping up if your childcare falls through. But for many of those things you describe, it's just a case of gritting your teeth and getting through it.
I think it would be different if your DH was employed and could take parental leave then absolutely, use that to provide as much support to you as is possible/you need.
But he's self-employed and you've said there would be a risk of going under. You say that other than a small contribution to childcare, he pays for everything.
I also think that if you're unwell you don't use resting time to "catch up on housework". If you can do that, then your DH doesn't need to be taking time off work. Not when doing so could jeopardise the financial security of your family.
I threw up all the way through my pregnancy (twins) and it was utterly, utterly miserable so you really do have my sympathy. I hope the morning sickness passes soon and you get that second trimester bounce. That's undoubtedly making you feel much worse. But you spending a few days on the sofa with a snotty nose and a bit of a cough doesn't seem to warrant your DH putting your income and ergo your financial security at risk.
One thing I would say, however. I'm also ND and it's easier for me when things are clearer. It might be helpful if you work out exactly when you need your DH to take time off work, and when you can muddle through. With us ND folk, ambiguity is difficult and having clearer expectations can help us to meet them more effectively. If you're doing housework when your DH is giving you time to rest - even if you've dragged your arse around the house to do it - that's a very confusing and mixed message.
Also, hyper focus can be a problem for us. That means when we're presented with a problem, we focus excessively on it and find it very difficult to be distracted from that. If your DH is currently focussing on making sure the business is stable and performing, he will find it even harder to take a few days off. That's not an excuse, but it means that you need to be even clearer about when you need him to help - and to communicate that assertively. Just be sure you're considering the wider picture and are being reasonable - if he'd taken time off for all those illnesses you've listed, then I'm sure any business would struggle.
Hope you're all feeling better soon.